Showing papers on "Fourier transform published in 1994"

The fractional Fourier transform and time-frequency representations

Abstract: The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >

Generating surrogate data for time series with several simultaneously measured variables.

Abstract: We propose an extension to multivariate time series of the phase-randomized Fourier-transform algorithm for generating surrogate data. Such surrogate data sets must mimic not only the autocorrelations of each of the variables in the original data set, they must mimic the cross correlations between all the variables as well. The method is applied both to a simulated example (the three components of the Lorentz equations) and to data from a multichannel electroencephalogram.

Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

Abstract: A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.

Integral Geometry of Tensor Fields

Abstract: Introduction: the problem of determining a metric by its hodograph and a linearization of the problem the kinetic equation in a Riemannian manifold. Part 1 The ray transform of symmetric tensor fields on Euclidean space: the ray transform and its relationship to the Fourier transform description of the kernel of the ray transform in the smooth case equivalence of the first two statements of theorem 2.2.1 in the case n=2 proof of theorem 2.2.2. the ray transform of a field-distribution decomposition of a tensor field into potential and solenoidal parts a theorem on the tangent component a theorem on conjugate tensor fields on the sphere primality of the ideal ([x]2, ) description of the image of the ray transform integral moments of the function I f inversion formulas for the ray transform proof of theorem 2.12.1 inversion of the ray transform on the space of field-distributions the Plancherel formula for the ray transform applications of the ray transform to an inverse problem of photoelasticity further results. Part 2 Some questions of tensor analysis. Part 3 The ray transform on a Riemannian manifold. Part 4 The transverse ray transform. Part 5 The truncated transverse ray transform. Part 6 The mixed ray transform. Part 7 The exponential ray transform (Part contents)

A fast numerical method for computing the linear and nonlinear mechanical properties of composites

Abstract: This Note is devoted to a new iterative algorithm to compute the local and overall response of a composite from images of its (complex) microstructure. The elastic problem for a heterogeneous material is formulated with the help of a homogeneous reference medium and written under the form of a periodic Lippman-Schwinger equation. Using the fact that the Green's function of the pertinent operator is known explicitely in Fourier space, this equation is solved iteratively.The method is extended to the case where the individual constituents are elastic-plastic Von Mises materials with isotropic hardening

Two-dimensional imaging

Abstract: 1. Introduction. 2. The Image Plane. 3. Two Dimensional Impulses. 4. The Two Dimensional Fourier Transform. 5. Two Dimensional Convolution. 6. The Convolution Theorem. 7. Sampling and Interpolation in Two Dimensions. 8. Digital Operations. 9. Rotational Symmetry and the Two Dimensional Fourier Transform. 10. Imaging by Convolution. 11. Diffraction Theory of Sensors. 12. Indirect Imaging and Interferometry. 13. Restoration of Images. 14. The Projection-Slice Theorem. 15. Computed Tomography. 16. Synthetic Aperture Radar. 17. Random Images and Fractals. Index.

Estimating frequency by interpolation using Fourier coefficients

Abstract: The periodogram of a time series that contains a sinusoidal component provides a crude estimate of its frequency parameter, the maximizer over the Fourier frequencies being within O(T/sup -1/) of the frequency as the sample size T increases. In the paper, a technique for obtaining an estimator that has root mean square error of order T/sup -3/2/ is presented, which involves only the Fourier components of the time series at three frequencies, The asymptotic variance of the estimator varies between, roughly, the asymptotic variance of the maximizer of the periodogram over all frequencies (the Cramer-Rao lower bound) and three times this variance. The advantage of the new estimator is its computational simplicity. >

Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison

Abstract: A detailed comparison of the original Gerchberg-Saxton and the Yang-Gu algorithms for the reconstruction of model images from two intensity measurements in a nonunitary transform system is presented. The Yang-Gu algorithm is a generalization of the Gerchberg-Saxton algorithm and is effective in solving the general amplitude-phase-retrieval problem in any linear unitary or nonunitary transform system. For a unitary transform system the Yang-Gu algorithm is identical to the Gerchberg-Saxton algorithm. The reconstruction of images from data corrupted with random noise is also investigated. The simulation results show that the Yang-Gu algorithm is relatively insensitive to the presence of noise in data. In all cases studied the Yang-Gu algorithm always resulted in a highly accurate recovered phase.

Introduction to Fourier Analysis

Abstract: CONTINUOUS FOURIER ANALYSIS. Background. Fourier Series for Periodic Functions. The Fourier Integral. Fourier Transforms of Some Important Functions. The Method of Successive Differentiation. Frequency-Domain Analysis. Time-Domain Analysis. The Properties. The Sampling Theorems. DISCRETE FOURIER ANALYSIS. The Discrete Fourier Transform. Inside the Fast Fourier Transform. The Discrete Fourier Transform as an Estimator. The Errors in Fast Fourier Transform Estimation. The Four Kinds of Convolution. Emulating Dirac Deltas and Differentiation on the Fast Fourier Transform. THE USER'S MANUAL FOR THE ACCOMPANYING DISKS. Appendices. Answers to the Exercises. Index.

Least-squares two-dimensional phase unwrapping using FFT's

Abstract: A solution of the least-squares two-dimensional phase-unwrapping problem is presented that is simpler to understand and implement than previously published solutions. It extends the phase function to a periodic function using a mirror reflection, and the resulting equation is solved using the Fourier transform. >

Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )

Abstract: A complete characterization is given of closed shift-invariant subspaces of L2(Rd) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.

Wavelet Spectrum Analysis and Ocean Wind Waves

Abstract: Wavelet spectrum analysis is applied to a set of measured ocean wind waves data collected during the 1990 SWADE (Surface Wave Dynamics Experiment) program. The results reveal significantly new and previously unexplored insights on wave grouping parameterizations, phase relations during wind wave growth, and detecting wave breaking characteristics. These insights are due to the nature of the wavelet transform that would not be immediately evident using a traditional Fourier transform approach.

Fresnel diffraction and the fractional-order Fourier transform.

Abstract: Fractional-order Fourier transforms are adapted to the mathematical expression of Fresnel diffraction, just as the standard Fourier transform is adapted to Fraunhofer diffraction. The continuity of fractional Fourier transforms with respect to their orders corresponds to the continuity of wave propagation, and their composition is in accordance with the Huygens principle.

Relationships between the radon-wigner and fractional fourier transforms

Abstract: Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.

Learning Boolean Functions via the Fourier Transform

Abstract: The importance of using the “right” representation of a function in order to “approximate” it has been widely recognized. The Fourier Transform representation of a function is a classic representation which is widely used to approximate real functions (i.e. functions whose inputs are real numbers). However, the Fourier Transform representation for functions whose inputs are boolean has been far less studied. On the other hand it seems that the Fourier Transform representation can be used to learn many classes of boolean functions.

Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation.

Abstract: The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.

SO2 Absorption Cross-section Measurement in the UV using a Fourier Transform Spectrometer

Abstract: Absorption cross sections of SO2 have been recorded at 295 K at the resolutions of 2 and 16 cm−1. The 27000- to 40000-cm−1 spectral region has been investigated. The comparison with data available from the literature shows a good agreement between the different data sets (less than 5%). However, local discrepancies, for example at the peaks of absorption, can reach 20%.

Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform.

Abstract: Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a function's Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

Wavelet Analysis in Geophysics: An Introduction

Abstract: Wavelet analysis is a rapidly developing area of mathematical and application-oriented research in many disciplines of science and engineering. The wavelet transform is a localized transform in both space (time) and frequency, and this property can be advantageously used to extract information from a signal that is not possible to unravel with a Fourier or even windowed Fourier transform. Wavelet transforms originated in geophysics in early 1980's for the analysis of seismic signal. After a decade of significant mathematical formalism they are now also being exploited for the analysis of several other geophysical processes such as atmospheric turbulence, space-time rainfall, ocean wind waves, seafloor bathymetry, geologic layered structures, climate change, among others. Due to their unique properties, well suited for the analysis of natural phenomena, it is anticipated that there will be an explosion of wavelet applications in geophysics in the next several years. This chapter provides a basic introduction to wavelet transforms and their most important properties. The theory and applications of wavelets is developing very rapidly and we see this chapter only as a limited basic introduction to wavelets which we hope to be of help to the unfamiliar reader and provide motivation and references for further study.

Signals, Systems, and Transforms

Abstract: For sophomore/junior-level signals and systems courses in Electrical and Computer Engineering departments. This text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

Elastodynamic fundamental solutions for anisotropic solids

Abstract: SUMMARY 3-D and 2-D time-domain elastodynamic fundamental solutions (or Green's functions) for linearly elastic anisotropic materials are obtained by the Radon transform. Fundamental solutions in the frequency domain follow directly by a subsequent evaluation of the Fourier transforms of the time-domain solutions. The solutions are in the form of a surface integral over a unit sphere for 3-D cases and in the form of a contour integral over a unit circle for 2-D cases. The integrals have a simple structure that can be interpreted as a superposition of plane waves. The wavefields can be separated into singular and regular parts. The singular parts correspond to the elastostatic fundamental solutions. The regular parts are bounded continuous functions. The integrals have been evaluated numerically for several examples. The results presented in this paper have direct applications to the formulation of boundary-integral equations for bodies of anisotropic materials and for the subsequent solution of these equations by the boundary-element method.

Fingerprint controlled public key cryptographic system

Abstract: A public key cryptographic system is implemented as follows. In an enrolment apparatus, the unique number for use in generating the public key and private key of the system is generated by manipulation of fingerprint information of a subscriber. A filter is then generated which is a function both of the Fourier transform of the subscriber's fingerprint(s) and of the unique number. This filter is stored on a subscriber card. When the subscriber wishes to generate his public or private key, he inputs his card to a card reader of an apparatus and places his finger(s) on a fingerprint input. The apparatus generates an optical Fourier transform from the fingerprint input. The Fourier transform signal is incident on to a spatial light modulator programmed with the filter information from the card. An inverse transform is generated from the filtered signal and this is used to regenerate the unique number. The apparatus also has a subsystem for utilizing the private key to decrypt an input encrypted message.

An adaptive estimation of periodic signals using a Fourier linear combiner

Abstract: Presents an adaptive algorithm for estimating from noisy observations, periodic signals of known period subject to transient disturbances. The estimator is based on the LMS algorithm and works by tracking the Fourier coefficients of the data. The estimator is analyzed for convergence, noise misadjustment and lag misadjustment for signals with both time invariant and time variant parameters. The analysis is greatly facilitated by a change of variable that results in a time invariant difference equation. At sufficiently small values of the LMS step size, the system is shown to exhibit decoupling with each Fourier component converging independently and uniformly. Detection of rapid transients in data with low signal to noise ratio can be improved by using larger step sizes for more prominent components of the estimated signal. An application of the Fourier estimator to estimation of brain evoked responses is included. >

A Diffuse Reflectance Infrared Fourier Transform Spectroscopic Study of the Surface Reaction of NaCl with Gaseous NO2 and HNO3

Abstract: The heterogeneous reactions of gaseous NO 2 and HNO 3 [(2-29)×10 14 molecules cm -3 ] with particles of NaCl in the 1-5-μm size range at 298 K have been followed in real time using diffuse reflectance infrared Fourier transform spectrometry (DRIFTS) to obtain kinetic and mechanistic data. Both the NO 2 and HNO 3 reactions gave an identical sequence of absorption bands attributed to nitrate ions, the expected product of these reactions: (1) HNO 3(g) +NaCl (s) →NaNO 3(s) +HCl (g) and (2) 2NO 2(g) +NaCl (s) → NaNO 3(s) +ClNO (g)

Temperature-Insensitive Near-Infrared Spectroscopic Measurement of Glucose in Aqueous Solutions:

Abstract: A digital Fourier filter is combined with partial least-squares (PLS) regression to generate a calibration model for glucose that is insensitive to sample temperature. This model is initially created by using spectra collected over the 5000 to 4000 cm-1 spectral range with samples maintained at 37°C. The analytical utility of the model is evaluated by judging the ability to determine glucose concentrations from a set of prediction spectra. Absorption spectra in this prediction set are obtained by ratioing single-beam spectra collected from solutions at temperatures ranging from 32 to 41°C to reference spectra collected at 37°C. The temperature sensitivity of the underlying water absorption bands creates large baseline variations in prediction spectra that are effectively eliminated by the Fourier filtering step. The best model provides a mean standard error of prediction across temperatures of 0.14 mM (2.52 mg/dL). The benefits of the Fourier filtering step are established, and critical experimental parameters, such as number of PLS factors, mean and standard deviation for the Gaussian shaped Fourier filter, and spectral range, are considered.

Wavelet localization of the Radon transform

Abstract: The authors develop an algorithm which significantly reduces radiation exposure in X-ray tomography, when a local region of the body is to be imaged. The algorithm uses the properties of wavelets to essentially localize the Radon transform. This algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image, not the image module the nullspace of the local tomography operator, or the Lambda transform of the image. This is possible because the authors do not truly invert the interior Radon transform, but rather sample the Radon transform sparsely away from the local region of interest. Much attention in the field has been directed towards localized tomography. The authors believe that this technique represents a significant contribution towards this effort. >

Structure solution by minimal‐function phase refinement and Fourier filtering. II. Implementation and applications

Abstract: The minimal function, R(ϕ), has been used to provide the basis for a new computer-intensive direct-methods procedure that shows potential for providing fully automatic routine solutions for structures in the 200–400 atom range. This procedure, which has been called shake-and-bake, is an iterative process in which real-space filtering is alternated with phase refinement using a technique that reduces the value of R(ϕ). It has been successfully tested using experimental data for a dozen known structures ranging in size from 25 to 317 atoms and crystallizing in a variety of space groups. The details of this procedure, the parameters used and the results of these applications are described.