Showing papers on "Fourier transform published in 1985"

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform

Abstract: This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography

Abstract: The Fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the Cartesian grid required for fast numerical Fourier inversion. Although the Fourier inversion method is recognized as being computationally faster than the back-projection method for parallel ray projection data, the artifacts resulting from inaccurate interpolation have generally limited application of the method. This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels. The method has potential for application to other measurement geometries such as fan-beam projections and diffraction tomography and NMR imaging.

Digital filters and signal processing

Abstract: 1 Introduction: Terminology and Motivation.- 2 Discrete-Time Signals and Systems.- 2.0 Introduction.- 2.1 Discrete-Time Signals, or Sequences.- 2.2 Discrete-Time Systems and Filters.- 2.3 Stability and Causality.- Problems.- 3 The z Transform.- 3.0 Introduction.- 3.1 Definition of the z Transform.- 3.2 Inverse z Transform.- 3.3 Inverse z Transform for Causal Sequences.- 3.4 Properties of the z Transform.- Problems.- 4 Input/Output Relationships.- 4.0 Introduction.- 4.1 System Function and Frequency Response.- 4.2 Difference Equations.- 4.3 Geometric Evaluations of H(z) and H'(?).- 4.4 State Variables.- Problems.- 5 Discrete-Time Networks.- 5.0 Introduction.- 5.1 Flow Graph Properties.- 5.2 Network Structures.- 5.3 Properties of Network Coefficients.- 5.4 Special Discrete-Time Networks.- Problems.- 6 Sampling Continuous-Time Signals.- 6.0 Introduction.- 6.1 Fourier Transform Relationships.- 6.2 Discrete-Time Fourier Transform.- 6.3 Laplace Transform Relationships.- 6.4 Prefilters, Postfilters and D/A Converters.- Problems.- 7 Discrete Fourier Transform.- 7.0 Introduction.- 7.1 Derivation and Properties of the DFT.- 7.2 Zero Padding.- 7.3 Windows in Spectrum Analysis.- 7.4 FFT Algorithms.- 7.5 Prime-Factor FFT's.- 7.6 Periodogram.- Problems.- 8 IIR Filter Design by Transformation.- 8.0 Introduction.- 8.1 Classical Filter Designs.- 8.2 Impulse-Invariant Transformation.- 8.3 Bilinear Transformation.- 8.4 Spectral Transformation.- Problems.- 9 FIR Filter Design Techniques.- 9.0 Introduction.- 9.1 Window-Function Technique.- 9.2 Frequency-Sampling Technique.- 9.3 Equiripple Designs.- Problems.- 10 Filter Design by Modeling.- 10.0 Introduction.- 10.1 Autoregressive (all-pole) Filters.- 10.2 Moving-Average (all-zero) Filters.- 10.3 ARMA (pole/zero) Filters.- 10.4 Lattice Structures.- 10.5 Spectrum Analysis by Modeling.- Problems.- 11 Quantization Effects.- 11.0 Introduction.- 11.1 Coefficient Quantization.- 11.2 Signal Quantization.- 11.3 Dynamic Range and Scaling.- 11.4 Parallel and Cascade Forms.- 11.5 Limit-Cycle Oscillations.- 11.6 State-Space Structures.- Problems.- 12 Digital Filter Implementation.- 12.0 Introduction.- 12.1 Bit-Serial Arithmetic and VLSI.- 12.2 Distributed Arithmetic.- 12.3 Block IIR Implementations.- Problems.- 13 Filter and Systems Examples.- 13.0 Introduction.- 13.1 Interpolation and Decimation.- 13.2 Hilbert Transformation.- 13.3 Digital Oscillators and Synthesizers.- 13.4 Speech Synthesis.- 13.5 Cepstrum.- Problems.- Answers to Selected Problems.- References.

Circuits, Signals, and Systems

Abstract: These twenty lectures have been developed and refined by Professor Siebert during the more than two decades he has been teaching introductory Signals and Systems courses at MIT. The lectures are designed to pursue a variety of goals in parallel: to familiarize students with the properties of a fundamental set of analytical tools; to show how these tools can be applied to help understand many important concepts and devices in modern communication and control engineering practice; to explore some of the mathematical issues behind the powers and limitations of these tools; and to begin the development of the vocabulary and grammar, common images and metaphors, of a general language of signal and system theory.Although broadly organized as a series of lectures, many more topics and examples (as well as a large set of unusual problems and laboratory exercises) are included in the book than would be presented orally. Extensive use is made throughout of knowledge acquired in early courses in elementary electrical and electronic circuits and differential equations.Contents: Review of the "classical" formulation and solution of dynamic equations for simple electrical circuits; The unilateral Laplace transform and its applications; System functions; Poles and zeros; Interconnected systems and feedback; The dynamics of feedback systems; Discrete-time signals and linear difference equations; The unilateral Z-transform and its applications; The unit-sample response and discrete-time convolution; Convolutional representations of continuous-time systems; Impulses and the superposition integral; Frequency-domain methods for general LTI systems; Fourier series; Fourier transforms and Fourier's theorem; Sampling in time and frequency; Filters, real and ideal; Duration, rise-time and bandwidth relationships: The uncertainty principle; Bandpass operations and analog communication systems; Fourier transforms in discrete-time systems; Random Signals; Modern communication systems."Circuits, Signals, and Systems" is included in The MIT Press Series in Electrical Engineering and Computer Science, copublished with McGraw-Hill.

Projective imaging of pulsatile flow with magnetic resonance

Abstract: Noninvasive angiography with magnetic resonance is demonstrated. Signal arising in all structures except vessels that carry pulsatile flow is eliminated by means of velocity-dependent phase contrast, electrocardiographic gating, and image subtraction. Background structures become in effect transparent, enabling the three-dimensional vascular tree to be imaged by projection to a two-dimensional image plane. Image acquisition and processing are accomplished with entirely conventional two-dimensional Fourier transform magnetic resonance imaging techniques. When imaged at 0.6 tesla, vessels 1 to 2 millimeters in diameter are routinely detected in a 50-centimeter field of view with data acquisition times less than 15 minutes. Studies of normal and pathologic anatomy are illustrated in human subjects.

Measurement of spatial correlation functions using image processing techniques

Abstract: A procedure for using digital image processing techniques to measure the spatial correlation functions of composite heterogeneous materials is presented. Methods for eliminating undesirable biases and warping in digitized photographs are discussed. Fourier transform methods and array processor techniques for calculating the spatial correlation functions are treated. By introducing a minimal set of lattice‐commensurate triangles, a method of sorting and storing the values of three‐point correlation functions in a compact one‐dimensional array is developed. Examples are presented at each stage of the analysis using synthetic photographs of cross sections of a model random material (the penetrable sphere model) for which the analytical form of the spatial correlations functions is known. Although results depend somewhat on magnification and on relative volume fraction, it is found that photographs digitized with 512×512 pixels generally have sufficiently good statistics for most practical purposes. To illustrate the use of the correlation functions, bounds on conductivity for the penetrable sphere model are calculated with a general numerical scheme developed for treating the singular three‐dimensional integrals which must be evaluated.

A photon dose distribution model employing convolution calculations.

Abstract: A three-dimensional photon beam calculation is described which models the primary, first-scatter, and multiple-scatter dose components from first principles Three key features of the model are (1) a multiple-scatter calculation based on diffusion theory, (2) the demonstration of the modulation transfer function of the radiation dose transport process, and (3) the use of the finite fast Fourier transform to perform the required convolutions The results of calculations for cobalt-60 in a homogeneous phantom are used to verify the accuracy of the model

Material Characterization by the Inversion of V(z)

Abstract: Absfmet-It is demonstrated that the reflectance function R(@) of a liquid-solid interface can be obtained by inverting the complex V(z) data collected with an acoustic microscope. The inversion algorithm is based on a nonparaxial formulation of the V(z) integral, which establishes the Fourier transform relation between R(@) and V(z). Examples are given to show that with this measurement technique, the acoustic phase velocities of the propagating modes in the solid medium can easily be determined and material losses can be estimated. The same technique is also used for characterizing imaging performance of focused systems. Applications In thin-6lm measurement are also discussed.

Recursive methods for computing the Abel transform and its inverse

Abstract: The Abel transform and its inverse appear in a wide variety of problems in which it is necessary to reconstruct axisymetric functions from line-integral projections. We present a new family of algorithms, principally for Abel inversion, that are recursive and hence computationally efficient. The methods are based on a linear, space-variant, state-variable model of the Abel transform. The model is the basis for deterministic algorithms, applicable when data are noise free, and least-squares-estimation (Kalman filter) algorithms, which accommodate the noisy data case. Both one-pass (filtering) and two-pass (smoothing) estimators are considered. In computer simulations, the new algorithms compare favorably with previous methods for Abel inversion.

Uncertainty principle and uncertainty relations

Abstract: It is generally believed that the uncertainty relation Δq Δp≥1/2ħ, where Δq and Δp are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path of a particle in an interference experiment (such as the double-slit experiment). The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions. To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall widthW Ψ and the mean peak width wψ of a general wave function ψ and show that the productW Ψ w φ is bounded from below if φ is the Fourier transform of ψ. It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments.

Fourier Transform Ion Mobility Spectrometry

Abstract: Theorie du fonctionnement d'un spectrometre a deux portes. Interet pour la chromatographie de plasma, ou la detection

Linear predictive residual representation via non-iterative spectral reconstruction

Abstract: Method of encoding speech at medium to high bit rates while maintaining very high speech quality, as specifically directed to the coding of the linear predictive (LPC) residual signal using either its Fourier Transform magnitude or phase. In particular, the LPC residual of the speech signal is coded using minimum phase spectral reconstruction techniques by transforming the LPC residual signal in a manner approximately a minimum phase signal, and then applying spectral reconstruction techniques for representing the LPC residual signal by either its Fourier Transform magnitude or phase. The non-iterative spectral reconstruction technique is based upon cepstral coefficients through which the magnitude and phase of a minimum phase signal are related. The LPC residual as reconstructed and regenerated is used as an excitation signal to a LPC synthesis filter in the generation of analog speech signals via speech synthesis from which audible speech may be produced.

On the restriction of the Fourier transform to curves: endpoint results and the degenerate case

Abstract: For smooth curves F in Rn with certain curvature properties it is shown that the composition of the Fourier transform in Rn followed by restriction to r defines a bounded operator from LP(Rn) to Lq(F) for certain p, q. The curvature hypotheses are the weakest under which this could hold, and p is optimal for a range of q. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.

Description of the absorption spectrum of iodine recorded by means of Fourier Transform Spectroscopy : the (B-X) system

Abstract: An in extenso analysis of the (B-X) I2 iodine absorption spectrum recorded by means of Fourier Transform Spectroscopy is presented. It is shown that the 100 000 recorded transitions covering the 11 000-20 040 cm-1 range and published in several Atlases may be recalculated by means of only 46 constants : 45 are Dunham coefficients describing the vibrational and rotational constants of both X state (up to v" = 19) and B state (up to v' = 80, situated only at 1.6 cm-1 from the dissociation limit of the B state), and one empirical scaling factor which takes account of neglected centrifugal constants higher than Mv. The overall standard error between computed and measured wavenumbers is equal to 0.002 cm -1 in agreement with the differences of numerous independent absolute wavenumbers and the computed ones.

A simplified approach for modulation transfer function determinations in computed tomography.

Abstract: In order to determine the modulation transfer functions (MTF's) for x-ray computed tomography (CT) scanners, a measurement must be performed to obtain either the point spread function (PSF) or the line spread function (LSF). Thereafter, the usual procedure is to interpolate between the measured points and to determine the Fourier transforms numerically in order to obtain the MTF. Since this must usually be done many times to evaluate various reconstruction kernels and scan modalities, the process is tedious. Fortunately, it can be greatly simplified by utilizing a mathematical function to describe the PSF or LSF. Measured data for five CT scanners indicates that the PSF can usually be described by a Gaussian function. Hence, the MTF can be written in a generalized form eliminating the necessity of performing Fourier transformations each time. The MTF is determined directly from a single performance characteristic related to the full width at half maximum. The accuracy of the approach is compared with detailed MTF calculations for five CT scanners and it is shown to agree favorably with this data.

Weakly convergent expansions of a plane wave and their use in Fourier integrals

Abstract: The Fourier transform of an irreducible spherical tensor is normally computed with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. The angular integrations are then trivial. However, the remaining radial integral containing a spherical Bessel function may be so complicated that the applicability of Fourier transformation is severely restricted. As an alternative, the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested. The weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert space L2(R3) or the Sobolev space W(1)2(R3). Accordingly, these weakly convergent expansions may be viewed as distributions that are defined on either L2(R3) or W(1)2(R3). The properties of some complete orthonormal se...

A New Time Domain Technique for Velocity Measurements Using Doppler Ultrasound

Abstract: A new technique for determining the Doppler frequency shift in a phase-coherent pulsed Doppler system is presented. In the new approach, the Doppler frequency shift is given directly in the time domain in terms of the measured I and Q components of the measured Doppler signal. The algorithm is based on an expression for the instantaneous rate of change of phase which separates rapidly varying from slowly varying terms. It permits noise smoothing in each term separately. Since the technique relies solely on signal processing in the time domain, it is significantly simpler to implement than the classic Fourier transform approach. In addition, the algorithm can be shown to give rigorously accurate values for instantaneous frequency and outperform the Fourier transform approach in poor signal-to-noise environments. Experimental results are presented which confirm the superiority of the new domain technique.

Magnetic resonance imaging the velocity vector components of fluid flow.

Abstract: Encoding the Precession phase angle of proton nuclei for Fourier analysis has produced accurate measurement of fluid velocity vector components by MRI. A Pair of identical gradient pulses separated in time by exactly ½ TE, are used to linearly encode the phase of flow velocity vector components without changing the phase of stationary nuclei, Two-dimensional Fourier transformation of signals gave velocity density images of laminar flow in angled tubes which were in agreement with the laws of vector addition. These Velocity profile images provide a quantitative method for the investigation of fluid dynamics and hemodynamics.© 1985 Academic Press,Inc.

Radial Fourier Multipliers and Associated Maximal Functions

Abstract: Publisher Summary This chapter shows how a certain square function may be used to obtain “general” multiplier and maximal multiplier theorems for radial Fourier multipliers. The multiplier theorem extends to redial multipliers of R2 that are not smooth away from a one dimensional “singularity” and the maximal theorem generalizes the results of concerning almost everywhere convergence of Bochner–Riesz means on R2 to a wider class of functions, as well as providing a unified approach to certain other operators associated to maximal and pointed convergence problem including Stein's spherical maximal function, and the solution operator to the linearized Schrodinger equation.

An introduction to Maslov's asymptotic method

Abstract: Summary. Familiar concepts such as asymptotic ray theory and geometrical spreading are now recognized as an asymptotic form of a more general asymptotic solution to the non-separable wave equation. In seismology, the name Maslov asymptotic theory has been attached to this solution. In its simplest form, it may be thought of as a justification of disc-ray theory and it can be reduced to the WKBJ seismogram. It is a uniformly valid asymptotic solution, though. The method involves properties of the wavefronts and ray paths of the wave equation which have been established for over a century. The integral operators which build on these properties have been investigated only comparatively recently. These operators are introduced very simply by appealing to the asymptotic Fourier transform of Ziolkowski & Deschamps. This leads quite naturally to the result that phase functions in different domains of the spatial Fourier transform are related by a Legendre transformation. The amplitude transformation can also be inferred by this method. Liouville's theorem (the incompressibility of a phase space of position and slowness) ensures that it is always possible to obtain a uniformly asymptotic solution. This theorem can be derived by methods familiar to seismologists and which do not rely on the traditional formalism of classical mechanics. It can also be derived from the sympletic property of the equations of geometrical spreading and canonical transformations in general. The symplectic property plays a central role in the theory of high-frequency beams in inhomogeneous media.

Fourier methods with extended stability intervals for the korteweg-de vries equation.

Abstract: A full leap frog Fourier method for integrating the Korteweg–de Vries (KdV) equation $u_t + uu_x - \varepsilon u_{xxx} = 0$ results in an $O(N^{ - 3} )$ stability constraint on the time step, where N is the number of Fourier modes used. This stability limit is much more restrictive than the accuracy limit for many applications.In this paper we propose a method for which the stability limit is extended by treating the linear dispersive $u_{xxx} $ term implicitly. Thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit. The implicit method is implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear $uu_x $ term by leap frog. A second method we propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration. A linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that ...

The Fourier series operational matrix of integration

Abstract: A general expression of the Fourier operational matrix of integration P is derived which is analogous to that previously derived for other types of orthogonal functions such as Walsh, block-pulse, Laguerre, Legendre and Chebyshev. This matrix P may be used to solve problems like identification, analysis and optimal control.