Results 1  10
of
104
The Chirplet Transform: Physical Considerations
, 1995
"... We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timef ..."
Abstract

Cited by 49 (3 self)
 Add to MetaCart
We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timefrequency plane and the timescale plane as twodimensional subspaces. The parameter space contains a "timefrequencyscale volume ", and thus encompasses both the shorttime Fourier transform (as a slice along the time and frequency axes), and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shearintime (obtained through convolution with a qchirp) and shearin frequency (obtained through multiplication by a qchirp). Signals in this multidimensional space can be obtained by a new transform which we call the "qchirplet transform", or simply the "chiplet transform". ...
Wave propagation using bases for bandlimited functions
 Wave Motion
, 2005
"... We develop a twodimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave eq ..."
Abstract

Cited by 27 (4 self)
 Add to MetaCart
(Show Context)
We develop a twodimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a firstorder system, we evolve the equation in time using the matrix exponential. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of onedimensional operators. We also use a partitioned lowrank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and computational time when compared with a fourthorder finite difference scheme in space and an explicit fourthorder Runge–Kutta solver in time.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
Spherical Slepian functions and the polar gap in geodesy
, 2006
"... Satellites mapping out the spatial variations of the gravitational or magnetic fields of the Earth or other planets ideally fly on polar orbits, uniformly covering the entire globe. Thus potential fields on the sphere are usually expressed in spherical harmonics, basis functions with global support. ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
Satellites mapping out the spatial variations of the gravitational or magnetic fields of the Earth or other planets ideally fly on polar orbits, uniformly covering the entire globe. Thus potential fields on the sphere are usually expressed in spherical harmonics, basis functions with global support. For various, especially engineering, reasons, however, inclined orbits are favorable. These leave a “polar gap”: an antipodal pair of axisymmetric polar caps, typically less than 10 ◦ in diameter, without any data coverage. Estimation of spherical harmonic field coefficients from an incompletely sampled sphere is prone to error, since the spherical harmonics are not orthogonal over the partial domain of the cut sphere. The historically somewhat neglected geodetic polar gap problem has been revived by, among others, Sneeuw & van Gelderen (1997), and recently,
MULTISTATIC IMAGING OF EXTENDED TARGETS ∗
"... Abstract. In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the nonmagnetic case and a highfrequency regime in the general case. Based on a highfrequency asymptotic analysis of the measurements, an algorithm for finding a good initial guess for the illuminated part of the inclusion is provided and its optimality is shown. We illustrate our main findings with a variety of numerical examples.
Uncertainty Principles in Hilbert Spaces
, 2002
"... In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities a ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.
E.: Integrating the Content and
 Process of Strategic MIS Planning with Competitive Strategy. Decision Sciences 22 (5
, 1991
"... We review here the recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations. The concept is introduced in successive steps through the studies of mapping of such computationally hard problems to the classical spin glass ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
We review here the recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations. The concept is introduced in successive steps through the studies of mapping of such computationally hard problems to the classical spin glass problems. The quantum spin glass problems arise with the introduction of quantum
A new friendly method of computing prolate spheroidal wave functions and wavelets
, 2005
"... Prolate spheroidal wave functions, because of their many remarkable properties leading to new applications, have recently experienced an upsurge of interest. They may be defined as eigenfunctions of either a differential operator or an integral operator (as observed by Slepian in the 1960’s). There ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
Prolate spheroidal wave functions, because of their many remarkable properties leading to new applications, have recently experienced an upsurge of interest. They may be defined as eigenfunctions of either a differential operator or an integral operator (as observed by Slepian in the 1960’s). There are various ways of calculating their values based on both approaches. The standard one uses an approximation based on Legendre polynomials, which, however, is valid only on a finite interval. An alternative, valid in a neighborhood of infinity, uses a Bessel function approximation. In this paper we present a new method based on an eigenvalue problem for a matrix operator equivalent to that of the integral operator. Its solution gives the values of these functions on the entire real line and is computationally more efficient. Key words: sampling theory, prolate spheroidal wave functions. 1
NonBoolean Descriptions for MindMatter Problems
"... A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmat ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmatter distinction. Such a description in terms of a locally Boolean but globally nonBoolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely nonBoolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a nonBoolean structure and can be encompassed into a single nonBoolean description. Tensed and tenseless time can be synchronized by holistic correlations. 1.
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
, 909
"... It is a wellknown fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access t ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
It is a wellknown fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are “spatiospectrally” concentrated, i.e. “localized ” in both domains at the same time. Here, we give a theoretical overview of one particular approach to this “concentration ” problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.