Showing papers on "Fourier transform published in 2000"

Spectral Methods in MATLAB

Abstract: Preface 1 Differentiation matrices 2 Unbounded grids: the semidiscrete Fourier transform 3 Periodic grids: the DFT and FFT 4 Smoothness and spectral accuracy 5 Polynomial interpolation and clustered grids 6 Chebyshev differentiation matrices 7 Boundary value problems 8 Chebyshev series and the FFT 9 Eigenvalues and pseudospectra 10 Time-stepping and stability regions 11 Polar coordinates 12 Integrals and quadrature formulas 13 More about boundary conditions 14 Fourth-order problems Afterword Bibliography Index

Foundations of Time-Frequency Analysis

Abstract: Time-frequency analysis is a modern branch of harmonic analysis. It com- prises all those parts of mathematics and its applications that use the struc- ture of translations and modulations (or time-frequency shifts) for the anal- ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym- metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen- tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita- tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori- entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.

Spatial filtering for zero-order and twin-image elimination in digital off-axis holography

Abstract: Off-axis holograms recorded with a CCD camera are numerically reconstructed with a calculation of scalar diffraction in the Fresnel approximation. We show that the zero order of diffraction and the twin image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. We show that this operation enhances the contrast of the reconstructed images and reduces the noise produced by parasitic reflections reaching the hologram plane with an incidence angle other than that of the object wave.

Cardiac motion tracking using cine harmonic phase (harp) magnetic resonance imaging

Abstract: The present invention relates to a method of measuring motion of an object such as a heart by magnetic resonance imaging. A pulse sequence is applied to spatially modulate a region of interest of the object and at least one first spectral peak is acquired from the Fourier domain of the spatially modulated object. The inverse Fourier transform information of the acquired first spectral-peaks is computed and a computed first harmonic phase image is determined from each spectral peak. The process is repeated to create a second harmonic phase image from each second spectral peak and the strain is determined from the first and second harmonic phase images. In a preferred embodiment, the method is employed to determine strain within the myocardium and to determine change in position of a point at two different times which may result in an increased distance or reduced distance. The method may be employed to determine the path of motion of a point through a sequence of tag images depicting movement of the heart. The method may be employed to determine circumferential strain and radial strain.

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

Abstract: We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, , of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G*(ω), in the Fourier frequency domain, and the stress relaxation modulus, Gr(t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing as a local power law. If the logarithmic slope of can be accurately determined, these estimates generally perform well at the frequency extremes.

The discrete fractional Fourier transform

Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

Protein docking using spherical polar Fourier correlations

Abstract: We present a new computational method of docking pairs of proteins by using spherical polar Fourier correlations to accelerate the search for candidate low-energy conformations. Interaction energies are estimated using a hydrophobic excluded volume model derived from the notion of "overlapping surface skins," augmented by a rigorous but "soft" model of electrostatic complementarity. This approach has several advantages over former three-dimensional grid-based fast Fourier transform (FFT) docking correlation methods even though there is no analogue to the FFT in a spherical polar representation. For example, a complete search over all six rigid-body degrees of freedom can be performed by rotating and translating only the initial expansion coefficients, many unfeasible orientations may be eliminated rapidly using only low-resolution terms, and the correlations are easily localized around known binding epitopes when this knowledge is available. Typical execution times on a single processor workstation range from 2 hours for a global search (5 x 10(8) trial orientations) to a few minutes for a local search (over 6 x 10(7) orientations). The method is illustrated with several domain dimer and enzyme-inhibitor complexes and 20 large antibody-antigen complexes, using both the bound and (when available) unbound subunits. The correct conformation of the complex is frequently identified when docking bound subunits, and a good docking orientation is ranked within the top 20 in 11 out of 18 cases when starting from unbound subunits. Proteins 2000;39:178-194.

Robust template matching for affine resistant image watermarks

Abstract: Digital watermarks have been proposed as a method for discouraging illicit copying and distribution of copyrighted material. This paper describes a method for the secure and robust copyright protection of digital images. We present an approach for embedding a digital watermark into an image using the Fourier transform. To this watermark is added a template in the Fourier transform domain to render the method robust against general linear transformations. We detail a new algorithm based on polar maps for the accurate and efficient recovery of the template in an image which has undergone a general affine transformation. We also present results which demonstrate the robustness of the method against some common image processing operations such as compression, rotation, scaling, and aspect ratio changes.

Wavelet packet feature extraction for vibration monitoring

Abstract: Condition monitoring of dynamic systems based on vibration signatures has generally relied upon Fourier-based analysis as a means of translating vibration signals in the time domain into the frequency domain. However, Fourier analysis provided a poor representation of signals well localized in time. In this case, it is difficult to detect and identify the signal pattern from the expansion coefficients because the information is diluted across the whole basis. The wavelet packet transform (WPT) is introduced as an alternative means of extracting time-frequency information from vibration signatures. The resulting WPT coefficients provide one with arbitrary time-frequency resolution of a signal. With the aid of statistical-based feature selection criteria, many of the feature components containing little discriminant information could be discarded, resulting in a feature subset having a reduced number of parameters without compromising the classification performance. The extracted reduced dimensional feature vector is then used as input to a neural network classifier. This significantly reduces the long training time that is often associated with the neural network classifier and improves its generalization capability.

Digital curvelet transform: strategy, implementation, and experiments

Abstract: Recently, Candes and Donoho introduced the curvelet transform, a new multiscale representation suited for objects which are smooth away from discontinuities across curves. Their proposal was intended for functions f defined on the continuum plane R2. In this paper, we consider the problem of realizing this transform for digital data. We describe a strategy for computing a digital curvelet transform, we describe a software environment, Curvelet 256, implementing this strategy in the case of 256 X 256 images, and we describe some experiments we have conducted using it. Examples are available for viewing by web browser.

Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography.

Abstract: A new method of measurement that essentially combines Fourier-domain optical coherence tomography with spectroscopy is introduced. By use of a windowed Fourier transform it is possible to obtain, in addition to the object structure, spectroscopic information such as the absorption properties of materials. The feasibility of this new method for performing depth-resolved spectroscopy is demonstrated with a glass filter plate. The results are compared with theoretically calculated spectra by use of the well-known spectral characteristics of the light source and the filter plate.

Time-frequency analysis of myoelectric signals during dynamic contractions: a comparative study

Abstract: Introduces nonstationary signal analysis methods to analyze the myoelectric (ME) signals during dynamic contractions by estimating the time-dependent spectral moments. The time-frequency analysis methods including the short-time Fourier transform, the Wigner-Ville distribution, the Choi-Williams distribution, and the continuous wavelet transform were compared for estimation accuracy and precision on synthesized and real ME signals. It is found that the estimates provided by the continuous wavelet transform have better accuracy and precision than those obtained with the other time-frequency analysis methods on simulated data sets. In addition, ME signals from four subjects during three different tests (maximum static voluntary contraction, ramp contraction, and repeated isokinetic contractions) were also examined.

Effect of a crystallite size distribution on X‐ray diffraction line profiles and whole‐powder‐pattern fitting

Abstract: A distribution of crystallite size reduces the width of a powder diffraction line profile, relative to that for a single crystallite, and lengthens its tails. It is shown that estimates of size from the integral breadth or Fourier methods differ from the arithmetic mean of the distribution by an amount which depends on its dispersion. It is also shown that the form of `size' line profiles for a unimodal distribution is generally not Lorentzian. A powder pattern can be simulated for a given distribution of sizes, if it is assumed that on average the crystallites have a regular shape, and this can then be compared with experimental data to give refined parameters defining the distribution. Unlike `traditional' methods of line-profile analysis, this entirely physical approach can be applied to powder patterns with severe overlap of reflections, as is demonstrated by using data for nanocrystalline ceria. The procedure is compared with alternative powder-pattern fitting methods, by using pseudo-Voigt and Pearson VII functions to model individual line profiles, and with transmission electron microscopy (TEM) data.

A Method for Measuring Heteronuclear (1H−13C) Distances in High Speed MAS NMR

Abstract: Magic angle spinning (MAS) NMR structure determination is rapidly developing. We demonstrate a method to determine 1H−13C distances r CH with high precision from Lee−Goldburg cross-polarization (LG-CP) with fast MAS and continuous LG decoupling on uniformly 13C-enriched tyrosine·HCl. The sequence is γ-encoded, and 1H−13C spin-pair interactions are predominantly responsible for the polarization transfer while proton spin diffusion is prevented. When the CP amplitudes are set to a sideband of the Hartmann−Hahn match condition, the LG-CP signal builds up in an oscillatory manner, reflecting coherent heteronuclear transfer. Its Fourier transform yields an effective 13C frequency response that is very sensitive to the surrounding protons. This 13C spectrum can be reproduced in detail with MAS Floquet simulations of the spin cluster, based on the positions of the nuclei from the neutron diffraction structure. It is symmetric around ω = 0 and yields two well-resolved maxima. Measurement of CH distances is straig...

Closed-form discrete fractional and affine Fourier transforms

Abstract: The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.

Uncorrelated modes of the non-linear power spectrum

Abstract: Non-linear evolution causes the galaxy power spectrum to become broadly correlated over different wavenumbers. It is shown that pre-whitening the power spectrum – transforming the power spectrum in such a way that the noise covariance becomes proportional to the unit matrix – greatly narrows the covariance of power. The eigenfunctions of the covariance of the pre-whitened non-linear power spectrum provide a set of almost uncorrelated non-linear modes somewhat analogous to the Fourier modes of the power spectrum itself in the linear, Gaussian regime. These almost uncorrelated modes make it possible to construct a near-minimum variance estimator and Fisher matrix of the pre-whitened non-linear power spectrum analogous to the Feldman–Kaiser–Peacock (FKP) estimator of the linear power spectrum. The paper concludes with summary recipes, in gourmet, fine and fastfood versions, of how to measure the pre-whitened non-linear power spectrum from a galaxy survey in the FKP approximation. An appendix presents FFTLog, a code for taking the fast Fourier or Hankel transform of a periodic sequence of logarithmically spaced points, which proves useful in some of the manipulations.

Frequency-domain analysis of biomolecular sequences

Abstract: Motivation: Frequency-domain analysis of biomolecular sequences is hindered by their representation as strings of characters. If numerical values are assigned to each of these characters, then the resulting numerical sequences are readily amenable to digital signal processing. Results: We introduce new computational and visual tools for biomolecular sequences analysis. In particular, we provide an optimization procedure improving upon traditional Fourier analysis performance in distinguishing coding from noncoding regions in DNA sequences. We also show that the phase of a properly defined Fourier transform is a powerful predictor of the reading frame of protein coding regions. Resulting color maps help in visually identifying not only the existence of protein coding areas for both DNA strands, but also the coding direction and the reading frame for each of the exons. Furthermore, we demonstrate that color spectrograms can visually provide, in the form of local ‘texture’, significant information about biomolecular sequences, thus facilitating understanding of local nature, structure and function. Availability: All software for techniques described in this paper is available from the author upon request.

Image processing with the radial Hilbert transform: theory and experiments

Abstract: The Hilbert transform is useful for image processing because it can select which edges of an input image are enhanced and to what degree the edge enhancement occurs. However, the transform operation is one dimensional and is not applicable for arbitrarily shaped two-dimensional objects. We introduce a radially symmetric Hilbert transform that permits two-dimensional edge enhancement. We implement one-dimensional, two-dimensional, and radial Hilbert transforms with a programmable phase-only liquid-crystal spatial light modulator. Experimental results are presented.

Mock Fourier series and transforms associated with certain Cantor measures

Abstract: We extend the construction, originally due to Jorgensen and Pedersen, of spectral pairs {μ, Λ}, consisting of Cantor measures μ on ℝn and discrete sets Λ such that the exponentials with frequency in Λ form an orthonormal basis forL 2(μ). We give conditions under which these mock Fourier series expansions ofL 1(μ) functions converge in a weak sense, and for a dense set of continuous functions the convergence is uniform. We show how to construct spectral pairs (2(μ) of infinite Cantor measures with unbounded support such that $$\hat f(\lambda ) = \smallint e( - x \cdot \lambda )f(x)d\tilde \mu (x),$$ defined for a dense subset ofL 2(μ), extends to an isometry fromL 2(μ) ontoL 2(μ'), a kind of mock Fourier transform. Our constructions do not require self-similarity, but only a compatible product structure for the pairs. We also give an analogue of the Shannon Sampling Theorem to reconstruct a function whose Fourier transform is supported in the Cantor set associated with μ from its values on Λ.

Improved instantaneous frequency estimation using an adaptive short-time Fourier transform

Abstract: Instantaneous frequency estimation (IFE) arises in a variety of important applications, including FM demodulation. We present here a new time-frequency representation (TFR)-based approach to IFE based on an adaptive short-time Fourier transform (ASTFT). This TFR leads naturally to a type of short-term ML estimator of the IF. To further improve the performance, we apply a multistate hidden Markov model (HMM)-based post-estimation tracker. The end result is up to a 16-dB reduction in the threshold SNR over the frequency discriminator (FD) and an 8-dB improvement over the phase-locked loop (PLL) for a Rayleigh fading channel.

Solving the generalized indirect Fourier transformation (GIFT) by Boltzmann simplex simulated annealing (BSSA)

Abstract: The structure of colloidal particles can be studied with small-angle X-ray and neutron scattering (SAXS and SANS). In the case of randomly oriented systems, the indirect Fourier transformation (IFT) is a well established technique for the calculation of model-free real-space information. Interaction leads to an overlap of inter- and intraparticle scattering effects, preventing most detailed interpretations. The recently developed generalized indirect Fourier transformation (GIFT) technique allows these effects to be separated by assuming various models for the interaction, i.e. the so-called structure factors. The different analytical behaviour of these structure factors from that of the form factors, describing the intraparticle scattering, allows this separation. The mean-deviation surface is defined by the quality of the fit for different parameter sets of the structure factor. Its global minimum represents the solution. The former non-linear least-squares approach has proved to be inefficient and not very reliable. In this paper, the incorporation of the completely different Boltzmann simplex simulated annealing (BSSA) algorithm for finding the global minimum of the hypersurface is presented. This new method increases not only the calculation speed but also the reliability of the evaluation.

Double random fractional Fourier domain encoding for optical security

Abstract: We propose a new optical encryption technique using the fractional Fourier transform. In this method, the data are encrypted to a stationary white noise by two statistically independent random phase masks in fractional Fourier domains. To decrypt the data correctly, one needs to specify the fractional domains in which the input plane, encryp- tion plane, and output planes exist, in addition to the key used for en- cryption. The use of an anamorphic fractional Fourier transform for the encryption of two-dimensional data is also discussed. We suggest an optical implementation of the proposed idea. Results of a numerical simulation to analyze the performance of the proposed method are pre- sented. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)01811-0)

Waves on noncommutative space–time and gamma-ray bursts

Abstract: Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski space–time [xi, t]=ιλxi. A natural wave equation is derived and the associated phenomena of in vacuo dispersion are discussed. Assuming the deformation scale λ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative space–time, with the noncommutativity equivalent under Fourier transform to non-Abelianness of the "addition law" for momentum in Feynman diagrams. We argue that CPT violation effects of the type testable using the sensitive neutral-kaon system are to be expected in such a theory.

Optical image encryption based on multifractional Fourier transforms

Abstract: We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.

Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation

Abstract: The aim of this article is threefold. First, we use the FBI transform to set up a calculus for partial differential operators with nonsmooth coefficients. Next, this calculus allows us to prove Strichartz type estimates for the wave equation with nonsmooth coefficients. Finally, we use these Strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations. 1. Introduction. The FBI transform is, in a way, similar to the complex Fourier transform, in that for each function in Rn it provides a representation as a holomorphic function in R2n. However, in the case of the FBI transform we can identify naturally R2n with the phase space T*Rn. For a pseudodifferential operator with smooth symbol acting on functions in Rn one can produce by conjugation a corresponding formal series acting on func tions in R2n9 for which the first term is exactly the multiplication by the symbol. This series converges and has a nice representation in the Weyl calculus provided that the symbol of the operator is analytic. This is how the FBI transform has been used in the study of partial differential operators with analytic coefficients; see (12), (13), where this machinery is developed. Here, in a way, we do the opposite: we look at operators with nonsmooth coefficients, approximate the conjugated operator by a partial sum of the formal series, and prove error estimates. In the simplest case the approximate conjugate operator is exactly the multiplication by the symbol. This is also related to the Cordoba-Fefferman wave-packet transform in (3). In the third section we use the error estimates to reduce the Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients to weighted LP(Lq) ?> L2 estimates for the FBI transform. These, in turn, can be proved in the usual fashion, using appropriate oscillatory integral esti mates. In the last part of the article we explain how one can use the new Strichartz type estimates to improve the local theory for nonlinear hyperbolic equations beyond the classical setup. These results are not sharp and will be improved in subsequent articles.

Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

Abstract: Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.

Classification of power system disturbances using a fuzzy expert system and a Fourier linear combiner

Abstract: This paper presents a hybrid scheme using a Fourier linear combiner and a fuzzy expert system for the classification of transient disturbance waveforms in a power system. The captured voltage or current waveforms are passed through a Fourier linear combiner block to provide normalized peak amplitude and phase at every sampling instant. The normalized peak amplitude and computed slope of the waveforms are then passed on to a diagnostic module that computes the truth value of the signal combination and determines the class to which the waveform belongs. Several numerical tests have been conducted using EMTP programs to validate the disturbance waveform classification with the help of the new hybrid approach which is much simpler than the recently postulated ANN or wavelet based approaches.