Showing papers on "Fourier transform published in 1974"
TL;DR: In this article, a systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering, where the form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator.
Abstract: A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems
2,746 citations
TL;DR: The authors compare the Fourier algorithm and a search algorithm using a simulated phantom to speed the search algorithm by using fewer interactions leaves decreased resolution in the region just inside the skull which could mask a subdural hematoma.
Abstract: The Fourier reconstruction may be viewed simply in the spatial domain as the sum of each line integral times a weighting function of the distance from the line to the point of reconstruction A modified weighting function simultaneously achieves accuracy, simplicity, low computation time, as well as low sensitivity to noise Using a simulated phantom, the authors compare the Fourier algorithm and a search algorithm The search algorithm required 12 iterations to obtain a reconstruction of accuracy and resolution comparable to that of the Fourier reconstruction, and was more sensitive to noise To speed the search algorithm by using fewer interactions leaves decreased resolution in the region just inside the skull which could mask a subdural hematoma
2,100 citations
TL;DR: Estimation of the parameters of a single-frequency complex tone from a finite number of noisy discrete-time observations is discussed and appropriate Cramer-Rao bounds and maximum-likelihood estimation algorithms are derived.
Abstract: Estimation of the parameters of a single-frequency complex tone from a finite number of noisy discrete-time observations is discussed. The appropriate Cramer-Rao bounds and maximum-likelihood (MI.) estimation algorithms are derived. Some properties of the ML estimators are proved. The relationship of ML estimation to the discrete Fourier transform is exploited to obtain practical algorithms. The threshold effect of one algorithm is analyzed and compared to simulation results. Other simulation results verify other aspects of the analysis.
1,878 citations
TL;DR: The Fourier transform ion cyclotron resonance (FT-ICR) 1.5 mass spectrometer as discussed by the authors was developed for mass spectroscopy and it can be used to obtain the whole spectrum in a very short period of time.
957 citations
TL;DR: In this paper, the authors demonstrated that frequency-sweep excitation can provide the broad-band irradiation required to excite ion cyclotron resonances throughout any desired mass range.
355 citations
TL;DR: This chapter provides an overview of three-dimensional reconstruction from projections and reviews the algorithms that have been proposed to solve the reconstruction problem.
Abstract: Publisher Summary This chapter provides an overview of three-dimensional reconstruction from projections. The chapter reviews the algorithms that have been proposed to solve the reconstruction problem. The known reconstruction algorithms are classified into four categories—summation, the use of Fourier transform, analytic solution of the integral equations, and series expansion approaches. For each class of algorithms several points need to be considered—a general intuitive description, a precise mathematical description of a typical reconstruction method of the class, and a brief description of other methods in the class. All algorithms for reconstruction take as input the projection data, and all produce as output an estimate of the original structure based on the available data. The estimate varies from method to method. The relative performance of the various methods depends on the object and how the data are collected. The simplest algorithm for reconstruction is to estimate the density at a point by adding all the ray sums of the rays through that point. The Fourier method depends on transforming the projections into the Fourier space, where they define part of the Fourier transform of the whole object. Each projection may be shown to yield values on a central section of the Fourier space, which is a line or plane through the origin at an angle corresponding to the direction of the projection in real space.
314 citations
TL;DR: The CORDIC iteration is applied to several Fourier transform algorithms and a new, especially attractive FFT computer architecture is presented as an example of the utility of this technique.
Abstract: The CORDIC iteration is applied to several Fourier transform algorithms. The number of operations is found as a function of transform method and radix representation. Using these representations, several hardware configurations are examined for cost, speed, and complexity tradeoffs. A new, especially attractive FFT computer architecture is presented as an example of the utility of this technique. Compensated and modified CORDIC algorithms are also developed.
304 citations
TL;DR: In this paper, a model of elastic energy dissipation based on a memory mechanism with two degrees of freedom is applied to the problem of the determination of the response spectrum at the surface of a layer when the spectrum is given at the bottom.
Abstract: A model of elastic‐energy dissipation based on a memory mechanism with two degrees of freedom is applied to the problem of the determination of the response spectrum at the surface of a layer when the spectrum is given at the bottom. The reaction of the surface of the layer is obtained directly with the Laplace‐transform method. With the Fourier method, the amplification function is also found at the surface of the layer; it depends on the first power of the coefficient of viscosity—the Q−1 is proportional to a fractional power of the frequency. The reaction inside of the layer in the case of an infinite layer has also been obtained as function of time and in the Fourier space. It is verified that the amplification function (namely, the peak amplitude response of the free surface of the viscoelastic layer at a “resonant frequency”) depends strongly on the dissipation mechanism; a complete knowledge of the parameters of this mechanism could be of great help in the solution of many physical and engineering ...
242 citations
TL;DR: In this article, the Fourier relationship between slow passage and free induction decay is violated, in general, as the state of the system can not be described by a Boltzmann distribution of the populations nor by a unique spin temperature.
215 citations
TL;DR: In this article, a semi-polar representation of the spin operators is introduced, which makes possible, in the harmonic approximation, the definition of magnons for any wavelength at low temperature in one-dimensional or two-dimensional magnetic systems without long-range order, provided they are of the planar type.
Abstract: A « semi-polar » representation of the spin operators is introduced, which makes possible, in the harmonic approximation, the definition of magnons for any wavelength at low temperature in one-dimensional (= 1-D) or two-dimensional (= 2-D) magnetic systems without long-range order, provided they are of the « planar » type, i. e. they have an easy magnetization plane. The semi-polar representation is used to calculate the spin pair correlation function at low temperature. Its space-time Fourier transform (directly observable by neutron scattering) consists of a relatively broad peak due to spin fluctuations inside the easy plane, plus a narrower peak due to out-of-plane fluctuations. The intensity, width and lineshape of both peaks are calculated in both 1-D and 2-D cases for all momentum transfers, as well as the frequency shift as a function of temperature.
213 citations
TL;DR: In this article, a theory of Fresnel diffraction and Fourier optics results, involving merely coordinate transformations of the independent variables of the aperture ambiguity function, is derived in terms of certain moments of the amplitude of the incident wave.
Abstract: If a field g(x,z) satisfies the diffusion equation ∂2g/∂x2 + 2j k(∂g/∂z) = 0, then its ambiguity function χ(x,ν,z)=∫-∞∞g(η+x/2,z)g*(η-x/2,z)e-jνηd η satisfies the wave equation ν2(∂2χ/∂x2) − k2(∂2χ/∂z2) = 0. A theory of Fresnel diffraction and Fourier optics results, involving merely coordinate transformations of the independent variables of the aperture ambiguity function. As an application, a simple expression for the width of the diffracted beam is derived in terms of certain moments of the amplitude of the incident wave. The analysis is extended to signals crossing a layer of a random medium. At the exit plane, the field is partially coherent and it spreads as it propagates. The broadening of beam width due to the loss in coherence is related to the statistical properties of the layer.
01 Jul 1974
TL;DR: The feasibility of a method for the identification of a three-dimensional object from information contained in the boundary of its silhouettes is demonstrated and the method was tested for identification of four aircraft representing complex and nonconvex objects.
Abstract: The feasibility of a method for the identification of a three-dimensional object from information contained in the boundary of its silhouettes is demonstrated. A silhouette is characterized by parametric representation of its boundary curve in the complex plane. After normalization and transformation, a set of Fourier descriptors is derived for every silhouette. A minimum distance classifier uses the descriptors to identify the three-dimensional object and to estimate its position and attitude with respect to a known reference coordinate system. The method was tested for identification of four aircraft representing complex and nonconvex objects. Simulation results, quantitative and statistical, are presented.
TL;DR: The relation between instantaneous frequencies and Fourier components of a time-dependent signal is examined in this paper, where it is shown that, while the appropriately weighted first moments of the component frequency and the instantaneous frequency coincide, the same is not true of the higher moments.
Abstract: The relation between instantaneous frequencies and Fourier components of a time-dependent signal is examined. Examples are given in which large excursions of the instantaneous frequency do not correspond to any Fourier components, and in which Fourier components are not manifest in the instantaneous frequencies. It is shown that, while the appropriately weighted first moments of the component frequency and the instantaneous frequency coincide, the same is not true of the higher moments. There is no one-to-one relationship between the two frequencies which correspond to different measurements. These considerations may be important in the study of resonant interactions between atoms and electromagnetic fields.
TL;DR: In this article, the merits of different modulated molecular beam detection systems are discussed and the application of modern data handling methods described, and it is shown that by using time domain analysis together with fourier transform techniques, information on the kinetics of surface processes is obtained.
TL;DR: In this paper, the theoretical basis for the use of cross correlation to extract an undistorted spectrum from a rapid scan response (as first suggested by Dadok) is examined in detail.
TL;DR: Two hybrid coding systems utilizing a cascade of a unitary transformation and differential pulse code modulators (DPCM) systems are proposed and the performance of this system is compared to the performances of the two- dimensional DPCM and the standard two-dimensional transform encoders.
Abstract: Two hybrid coding systems utilizing a cascade of a unitary transformation and differential pulse code modulators (DPCM) systems are proposed. Both systems encode the transformed data by a bank of DPCM systems. The first system uses a one-dimensional transform of the data where the second one employs two-dimensional transformations. Theoretical results for Markov data and experimental results for a typical picture are presented for Hadamard, Fourier, cosine, slant, and the KarhunenLoeve transformations. The visual effects of channel error and also the impact of noisy channel on the performance of the hybrid system, measured in terms of the signal-to-noise ratio of the encoder, is examined and the performance of this system is compared to the performances of the two-dimensional DPCM and the standard two-dimensional transform encoders.
TL;DR: The convolution theorem of Fourier transformation theory provides an "exact" solution to the representation of frequency-dependent line constants of distributed-parameter transmission lines.
Abstract: The convolution theorem of Fourier transformation theory provides an "exact" solution to the representation of frequency-dependent line constants of distributed-parameter transmission lines. This paper emphasizes modelling aspects crucial to accurate and efficient numerical solution by digital computer, as part of a production electromagnetic transients program. Sample results simulating a field test are included.
01 Oct 1974
TL;DR: The basic Fourier transform relationship for coherently illuminated systems is developed and methods for constructing complex-valued spatial filters are described; these filters can be used to realize such operations as convolution or cross correlation, addition or subtraction, and differentiation or integration.
Abstract: The principles of communication theory were applied in the 1950's to optical imaging systems and to the analysis of images. Optical systems were analyzed in much the same way as linear systems (modulation transfer functions and channel capacities) and images were characterized in ways analogous to time signals (space-bandwidth products, spatial frequency content, etc.). Both coherently and incoherently illuminated optical systems can be treated using these concepts. Coherently illuminated systems are most useful for performing operation such as convolution, cross correlation, and spectral analysis because the Fourier transform of an optical signal physically exists and can, therefore, be measured or modified. The basic Fourier transform relationship for coherently illuminated systems is developed in this paper. It can be detected directly and used to estimate the distribution of spatial frequencies contained in the signal. Methods for constructing complex-valued spatial filters are described; these filters can be used to realize such operations as convolution or cross correlation, addition or subtraction, and differentiation or integration. Experimental results are given to illustrate the concepts and to susgest potential applications. To extend the range of applications, interface devices are needed to allow optical processing of two-dimensional raster-scanned time signals, wide bandwidth electrical signals, and incoherent optical signals. Interface devices are often needed to convert the output optical signal to an electrical signal for post-processing by a digital computer. For some applications, interface devices are needed to construct spatial filters in real time, so different operations can be performed on a given signal. The desired characteristics of these three interface devices and the current state of their development are briefly reviewed.
TL;DR: In this article, the authors used imperfect quadrature detection to remove reflections of lines in time-averaged Fourier transform NMR spectra obtained by imperfect quadRature detection, which can be either by 90° rf phase shifting of the excitation pulse together with simple data routing, or by phase and amplitude manipulations of the two imperfect free induction decays after completion of data accumulation.
TL;DR: In this paper, the authors show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η, ε can be realized in the Schrodinger representation η and −id/dε.
Abstract: Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x and p to a pair η and ζ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η, ζ can be realized in the Schrodinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square‐integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well‐known Bargmann transform is the special case when η and iζ are the harmonic oscillator raising and lowering operators. The Moshinsky‐Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.
01 Jan 1974
TL;DR: In this article, the duality theory of linear programs is derived using the Fourier method of elimination of variables, which can be used to derive both for finite and infinite linear programs.
Abstract: Fourier treated a system of linear inequalities by a method of elimination of variables. This method can be used to derive the duality theory of linear programming. Perhaps this furnishes the quickest proof both for finite and infinite linear programs. For numerical evaluation of a linear program, Fourier’s procedure is very cumbersome because a variable is eliminated by adding each pair of inequalities having coefficients of opposite sign. This introduces many redundant inequalities. However, modifications are possible which reduce the number of redundant inequalities generated. With these modifications the method of Fourier becomes a practical computational algorithm for a class of parametric linear programs.
TL;DR: In this article, the mutual coherence function of a laser beam propagating in turbulence with a modified Von Karman spectrum for the index-of-refraction fluctuations was computed using transport methods.
Abstract: Using transport methods, we have computed the mutual coherence function of a laser beam propagating in turbulence with a modified Von Karman spectrum for the index-of-refraction fluctuations, including a parametric study of the effect on the mutual coherence function of varying beam size, focal length, and the properties of the turbulence. The results are valid over nearly all propagation distances, unlike calculations employing the method of smooth perturbations. We have also studied the effect of a constant wind on the frequency spectrum of the signal, and have found that, over most practical path lengths, the spectral width is Δω=4.1 Vk06/5(Cn2)3/5z3/5, where V is the wind speed, k0 is the signal wave number, Cn2 is the strength of turbulence, and z is the path length in the turbulence.
TL;DR: In this article, the infrared spectrum of nitrous oxide has been measured and analyzed from 3000 cm −1 to 8200 cm − 1, and the spectra have been recorded with the Fourier transform interferometer of Laboratoire Aime Cotton.