Showing papers on "Fourier transform published in 2003"

Classical and modern Fourier analysis

Abstract: Prolegomena 1 L p Spaces and Interpolation 2 Maximal Functions, Fourier Transform, and Distributions 3 Fourier Analysis on the Torus 4 Singular Integrals of Convolution Type 5 Littlewood-Paley Theory and Multipliers 6 Smoothness and Function Spaces 7 BMO and Carleson Measures 8 Singular Integrals of Nonconvolution Type 9 Weighted Inequalities 10 Boundedness and Convergence of Fourier Integrals Bibliography Index of Notation Index

Two-dimensional femtosecond spectroscopy

Abstract: The simplest two-dimensional (2D) spectra show how excitation with one (variable) frequency affects the spectrum at all other frequencies, thus revealing the molecular connections between transitions. Femtosecond 2D Fourier transform (2D FT) spectra are more flexible and share some of the remarkable properties of their conceptual parent, 2D FT nuclear magnetic resonance. When 2D FT spectra are experimentally separated into real absorptive and imaginary refractive parts, the time resolution and frequency resolution can both reach the uncertainty limit set for each resonance by the sample itself. Coherent four-level contributions to the signal provide new molecular phase information, such as relative signs of transition dipoles. The nonlinear response can be picked apart by selecting a single coherence pathway (e.g., specifying the relative signs of energy level difference frequencies during different time intervals as in the photon echo). Because molecules are frozen on the femtosecond timescale, femtosecond 2D FT experiments can separate a distribution of instantaneous molecular environments and intramolecular geometries as inhomogeneous broadening. This review provides an introduction to two-dimensional Fourier transform experiments exploiting second- and third-order vibrational and electronic nonlinearities.

Coherent 2D IR Spectroscopy: Molecular Structure and Dynamics in Solution

Abstract: Two-dimensional infrared (2D IR) vibrational spectroscopy is an experimental tool for investigating molecular dynamics in solution on a picosecond time scale. We present experimental and theoretical methods for obtaining a 2D IR correlation spectrum and modeling the underlying microscopic information. Fourier transform 2D spectra are obtained from heterodyne-detected third-order nonlinear signals using a sequence of broad bandwidth femtosecond IR pulses. A 2D IR correlation spectrum with absorptive line shapes results from the addition of 2D rephasing and nonrephasing spectra, which sample conjugate frequencies during the initial evolution time period. The 2D IR spectrum contains peaks with different positions, signs, amplitudes, and line shapes characterizing the vibrational eigenstates of the system and their interactions with the surrounding bath. The positions of the peaks map the transition frequencies between the ground, singly, and doubly excited states of the system and thus describe the anharmoni...

Applications of Hilbert–Huang transform to non-stationary financial time series analysis

Abstract: A new method, the Hilbert–Huang Transform (HHT), developed initially for natural and engineering sciences has now been applied to financial data. The HHT method is specially developed for analysing non-linear and non-stationary data. The method consists of two parts: (1) the empirical mode decomposition (EMD), and (2) the Hilbert spectral analysis. The key part of the method is the first step, the EMD, with which any complicated data set can be decomposed into a finite and often small number of intrinsic mode functions (IMF). An IMF is defined here as any function having the same number of zero-crossing and extrema, and also having symmetric envelopes defined by the local maxima, and minima respectively. The IMF also thus admits well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to non-linear and non-stationary processes. With the Hilbert transform, the IMF yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy–frequency–time distribution, which we designate as the Hilbert Spectrum. Comparisons with Wavelet and Fourier analyses show the new method offers much better temporal and frequency resolutions. The EMD is also useful as a filter to extract variability of different scales. In the present application, HHT has been used to examine the changeability of the market, as a measure of volatility of the market. Published in 2003 by John Wiley & Sons, Ltd.

Fourier Analysis: An Introduction

Abstract: Foreword vii Preface xi Chapter 1. The Genesis of Fourier Analysis 1 Chapter 2. Basic Properties of Fourier Series 29 Chapter 3. Convergence of Fourier Series 69 Chapter 4. Some Applications of Fourier Series 100 Chapter 5. The Fourier Transform on R 129 Chapter 6. The Fourier Transform on R d 175 Chapter 7. Finite Fourier Analysis 218 Chapter 8. Dirichlet's Theorem 241 Appendix: Integration 281 Notes and References 299 Bibliography 301 Symbol Glossary 305

The S‐transform with windows of arbitrary and varying shape

Abstract: The S-transform is an invertible time-frequency spectral localization technique which combines elements of wavelet transforms and short-time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S-transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S-transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram. The S-transform is also used as a time-frequency filter; this helps in determining the sign of the P arrival.

Optical image encryption by random shifting in fractional Fourier domains

Abstract: A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined.

Fast phase unwrapping algorithm for interferometric applications.

Abstract: A wide range of interferometric techniques recover phase information that is mathematically wrapped on the interval (-π,π] . Obtaining the true unwrapped phase is a longstanding problem. We present an algorithm that solves the phase unwrapping problem, using a combination of Fourier techniques. The execution time for our algorithm is equivalent to the computation time required for performing eight fast Fourier transforms and is stable against noise and residues present in the wrapped phase. We have extended the algorithm to handle data of arbitrary size. We expect the state of the art of existing interferometric applications, including the possibility for real-time phase recovery, to benefit from our algorithm.

Latest views of the sparse Radon transform

Abstract: The Radon transform (RT) suffers from the typical problems of loss of resolution and aliasing that arise as a consequence of incomplete information, including limited aperture and discretization. Sparseness in the Radon domain is a valid and useful criterion for supplying this missing information, equivalent somehow to assuming smooth amplitude variation in the transition between known and unknown (missing) data. Applying this constraint while honoring the data can become a serious challenge for routine seismic processing because of the very limited processing time available, in general, per common midpoint. To develop methods that are robust, easy to use and flexible to adapt to different problems we have to pay attention to a variety of algorithms, operator design, and estimation of the hyperparameters that are responsible for the regularization of the solution. In this paper, we discuss fast implementations for several varieties of RT in the time and frequency domains. An iterative conjugate gradient algorithm with fast Fourier transform multiplication is used in all cases. To preserve the important property of iterative subspace methods of regularizing the solution by the number of iterations, the model weights are incorporated into the operators. This turns out to be of particular importance, and it can be understood in terms of the singular vectors of the weighted transform. The iterative algorithm is stopped according to a general cross validation criterion for subspaces. We apply this idea to several known implementations and compare results in order to better understand differences between, and merits of, these algorithms.

Persistent angular structure: new insights from diffusion magnetic resonance imaging data

Abstract: We determine a statistic called the (radially) persistent angular structure (PAS) from samples of the Fourier transform of a three-dimensional function. The method has applications in diffusion magnetic resonance imaging (MRI), which samples the Fourier transform of the probability density function of particle displacements. The PAS is then a representation of the relative mobility of particles in each direction. In PAS-MRI, we compute the PAS in each voxel of an image. This technique has biomedical applications, where it reveals the orientations of microstructural fibres, such as white-matter fibres in the brain. Scanner time is a significant factor in determining the amount of data available in clinical brain scans. Here, we use measurements acquired for diffusion-tensor MRI, which is a routine diffusion imaging technique, but extract richer information. In particular, PAS-MRI can resolve the orientations of crossing fibres. We test PAS-MRI on human brain data and on synthetic data. The human brain data set comes from a standard acquisition scheme for diffusion-tensor MRI in which the samples in each voxel lie on a sphere in Fourier space.

The FFT on a GPU

Abstract: The Fourier transform is a well known and widely used tool in many scientific and engineering fields. The Fourier transform is essential for many image processing techniques, including filtering, manipulation, correction, and compression. As such, the computer graphics community could benefit greatly from such a tool if it were part of the graphics pipeline. As of late, computer graphics hardware has become amazingly cheap, powerful, and flexible. This paper describes how to utilize the current generation of cards to perform the fast Fourier transform (FFT) directly on the cards. We demonstrate a system that can synthesize an image by conventional means, perform the FFT, filter the image, and finally apply the inverse FFT in well under 1 second for a 512 by 512 image. This work paves the way for performing complicated, real-time image processing as part of the rendering pipeline.

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes

Abstract: A method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are first decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identified using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and stiffness matrices of the structure. The effectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identified effectively using the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

Abstract: The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.

Optimal control theory for unitary transformations

Abstract: The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time-dependent Hamiltonian The inverse problem of finding the field that generates a specific unitary transformation is the subject of study The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space Optimal control theory is used to solve the inversion problem irrespective of the initial input state A unified formalism based on the Krotov method is developed leading to a different scheme The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ electronic state of ${\mathrm{Na}}_{2}$ Raman-like transitions through the $A{}^{1}{\ensuremath{\Sigma}}_{u}^{+}$ electronic state induce the transitions Light fields are found that are able to implement the Fourier transform within a picosecond time scale Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse Of the schemes studied, the square modulus scheme converges fastest A study of the implementation of the Q qubit Fourier transform in the ${\mathrm{Na}}_{2}$ molecule was carried out for up to five qubits The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized

Simple plane wave implementation for photonic crystal calculations

Abstract: A simple implementation of plane wave method is presented for modeling photonic crystals with arbitrary shaped ‘atoms’ The Fourier transform for a single ‘atom’ is first calculated either by analytical Fourier transform or numerical FFT, then the shift property is used to obtain the Fourier transform for any arbitrary supercell consisting of a finite number of ‘atoms’ To ensure accurate results, generally, two iterating processes including the plane wave iteration and grid resolution iteration must converge Analysis shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration It converges to the accurate results quickly using a small number of plane waves Coordinate conversion is used to treat non-orthogonal unit cell with non-regular ‘atom’ and then is treated by standard numerical FFT MATLAB source code for the implementation requires about less than 150 statements, and is freely available at http://wwwlionsoduedu/~sguox002

Ab Initio Molecular Dynamics Computation of the Infrared Spectrum of Aqueous Uracil

Abstract: Recent progress in the development of ab initio molecular dynamics methods for the computation of infrared absorption spectra in condensed molecular systems is reviewed and illustrated by a detailed account of an application to aqueous uracil. Similar to classical force field simulations, the spectrum is obtained as the Fourier transform of the polarization time autocorrelation function. The density functional methodology for the computation of electronic polarization in periodic supercells is briefly outlined, and also the effect of quantum corrections is discussed. The spectral patterns obtained for the model system in the 2000−1000 cm-1 domain are in good agreement with experiment. Comparing to the low-temperature vacuum spectrum computed by similar time-dependent methods, we found that the narrow amide bending band in a vacuum is spread out over a 500 cm-1 wide interval in solution with a substantially blue-shifted high-frequency end. The highest increase in frequency was found for N1−H1 bending. The ...

Fourier Analysis and Its Applications

Abstract: Introduction.- Preparations.- Laplace and Z Transforms.- Fourier Series.- L^2 Theory.- Separation of Variables.- Fourier Transforms.- Distributions.- Multi-Dimensional Fourier Analysis.- Appendix A: The ubiquitous convolution.- Appendix B: The Discrete Fourier Transform.- Appendix C: Formulae.- Appendix D: Answers to exercises.- Appendix E: Literature.

Full Spectrum Inversion of radio occultation signals

Abstract: [1] Temperature, pressure, and humidity profiles of the Earth's atmosphere can be derived through the radio occultation technique. This technique is based on the Doppler shift imposed, by the atmosphere, on a signal emitted from a GNSS satellite and received by a low orbiting satellite. The method is very accurate with a temperature accuracy of 1°K, when both frequencies in the GPS system are used. However, difficulties arise when the signal consists of multiple frequencies generated by multipath phenomena in the atmosphere. We demonstrate that, in general, it is possible to determine the arrival times of the different frequency components in a radio occultation signal simply as the derivatives of the phases of the conjugated Fourier spectrum of the entire occultation signal. Based on this property, a novel Full Spectrum Inversion technique for radio occultation sounding capable of disentangling multiple rays in multipath regions is presented. As the entire signal is used in the Fourier transform, a high spatial resolution in the Doppler frequency, and hence in the retrieved temperature, pressure, and humidity profiles, can be achieved. The method is conceptual and computational simple and thus easy to implement. The performance of the Full Spectrum Inversion is demonstrated by applying the technique to simulated signals generated by solving Helmholtz equation with use of the multiple phase-screen technique. Excellent agreement is found between computed bending angle profiles and corresponding solutions to the Abel integral.

Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors

Abstract: A Fourier modal method for analysing crossed anisotropic gratings is presented without any restrictions on the permittivity and permeability tensors of the medium in the grating region and the angle between the two periodic directions. A skew Cartesian coordinate system with skew angles both in and out of the grating plane is used in the mathematical derivation, which gives the formulation a greater generality. However, the facets of the crossed gratings are required to be parallel to the skew coordinate surfaces. Correct Fourier factorization of Maxwell's equations is carried out with the aid of two efficient operators that greatly simplify the otherwise complicated notation. Numerical curves are presented to demonstrate the convergence of the method, and numerical values are tabulated to provide reference data.

Short-time Fourier transform: two fundamental properties and an optimal implementation

Abstract: Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.

Introduction to the mathematics of medical imaging

Abstract: Preface to the second edition Preface How to use this nook Notational conventions 1. Measurements and modeling 2. Linear models and linear equations 3. A basic model for tomography 4. Introduction to the Fourier transform 5. Convolution 6. The radon transform 7. Introduction to Fourier series 8. Sampling 9. Filters 10. Implementing shift invariant filters 11. Reconstruction in X-ray tomography 12. Imaging artifacts in X-ray tomography 13. Algebraic reconstruction techniques 14. Magnetic resonance imaging 15. Probability and random variables 16. Applications of probability 17. Random processes A. Background material B. Basic analysis Bibliography Index.

Pilot-aided channel estimation for OFDM in wireless systems

Abstract: A method and apparatus for pilot-symbol aided channel estimation in a wireless digital communication system which transmits packets of N OFDM data blocks, each data block comprising a set of K orthogonal carrier frequencies. At the transmitter, pilot symbols are inserted into each data packet at known positions so as to occupy predetermined positions in the time-frequency space. At the receiver, the received signal is subject to a two-dimensional inverse Fourier transform, two-dimensional filtering and a two-dimensional Fourier transform to recover the pilot symbols so as to estimate the channel response.

UV divergence-free QFT on noncommutative plane

Abstract: We formulate noncommutative qauntum field theory in terms of fields defined as mean value over coherent states of the noncommutative plane. No -product is needed in this formulation and noncommutativity is carried by a modified Fourier transform of fields. As a result the theory is UV finite and the cutoff is provided by the noncommutative parameter θ.

nMoldyn: a program package for a neutron scattering oriented analysis of molecular dynamics simulations.

Abstract: We present a new implementation of the program nMoldyn, which has been developed for the computation and decomposition of neutron scattering intensities from Molecular Dynamics trajectories (Comp. Phys. Commun 1995, 91, 191-214). The new implementation extends the functionality of the original version, provides a much more convenient user interface (both graphical/interactive and batch), and can be used as a tool set for implementing new analysis modules. This was made possible by the use of a high-level language, Python, and of modern object-oriented programming techniques. The quantities that can be calculated by nMoldyn are the mean-square displacement, the velocity autocorrelation function as well as its Fourier transform (the density of states) and its memory function, the angular velocity autocorrelation function and its Fourier transform, the reorientational correlation function, and several functions specific to neutron scattering: the coherent and incoherent intermediate scattering functions with their Fourier transforms, the memory function of the coherent scattering function, and the elastic incoherent structure factor. The possibility to compute memory function is a new and powerful feature that allows to relate simulation results to theoretical studies.

Short-time fractional Fourier methods for the time-frequency representation of chirp signals.

Abstract: The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, a, related to chirp rate. The two short-time implementations differ in how the value of a is chosen. In the first, a global optimization procedure selects one value of a with reference to the entire signal. In the second, a values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.

Lectures on Harmonic Analysis

Abstract: The $L^1$ Fourier transform The Schwartz space Fourier inversion and the Plancherel theorem Some specifics, and $L^p$ for $p<2$ The uncertainty principle The stationary phase method The restriction problem Hausdorff measures Sets with maximal Fourier dimension and distance sets The Kakeya problem Recent work connected with the Kakeya problem Bibliography for Chapter 11 Historical notes Bibliography.

Seismic trace interpolation in the Fourier transform domain

Abstract: A data adaptive interpolation method is designed and applied in the Fourier transform domain (f‐k or f‐kx‐ky for spatially aliased data. The method makes use of fast Fourier transforms and their cyclic properties, thereby offering a significant cost advantage over other techniques that interpolate aliased data.The algorithm designs and applies interpolation operators in the f‐k (or f‐kx‐ky domain to fill zero traces inserted in the data in the t‐x (or t‐x‐y) domain at locations where interpolated traces are needed. The interpolation operator is designed by manipulating the lower frequency components of the stretched transforms of the original data. This operator is derived assuming that it is the same operator that fills periodically zeroed traces of the original data but at the lower frequencies, and corresponds to the f‐k (or f‐kx‐ky domain version of the well‐known f‐x (or f‐x‐y) domain trace interpolators.The method is applicable to 2D and 3D data recorded sparsely in a horizontal plane. The most comm...

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

Abstract: We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\mathbb{R}^d$ which may be written as $P(x)\exp (-\langle Ax, x\rangle)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x) \widehat{f}(y)$. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

Semiclassical Theory of Vibrational Energy Relaxation in the Condensed Phase

Abstract: This paper presents the first application of semiclassical methodology to the calculation of vibrational energy relaxation (VER) rate constants in condensed phase systems. The VER rate constant is treated within the framework of the Landau-Teller formula and is given in terms of the Fourier transform, at the vibrational frequency, of the force-force correlation function (FFCF). Due to the high frequency of most molecular vibrations, predictions based on the classical FFCF are often found to deviate by orders of magnitude from the experimentally observed values. In this paper, we employ a semiclassical approximation for the quantum-mechanical FFCF, that puts it in terms of a classical-like expression, where Wigner transforms replace the corresponding classical quantities. The multidimensional Wigner transform is performed via a novel implementation of the local harmonic approximation (LHA). The resulting expression for the FFCF is exact at t = 0, and converges to the correct classical limit when h → 0. Quantum effects are introduced via a nonclassical initial sampling of both positions and momenta, as well as by accounting for delocalization in the calculation of the force at t = 0. The application of the semiclassical method is reported for three model systems: (1) a vibrational mode coupled to a harmonic bath, with the coupling exponential in the bath coordinates; (2) a diatomic coupled to a short linear chain of helium atoms; (3) a "breathing sphere" diatomic in a two-dimensional monatomic Lennard-Jones liquid. Good agreement is found in all cases between the semiclassical predictions and the exact results, or their estimates. It is also found that the VER of highfrequency molecular vibrations is dominated by a purely quantum-mechanical term, which vanishes in the classical limit.