Showing papers on "Fourier transform published in 1969"

A guided tour of the fast Fourier transform

Abstract: For some time the Fourier transform has served as a bridge between the time domain and the frequency domain. It is now possible to go back and forth between waveform and spectrum with enough speed and economy to create a whole new range of applications for this classic mathematical device. This article is intended as a primer on the fast Fourier transform, which has revolutionized the digital processing of waveforms. The reader's attention is especially directed to the IEEE Transactions on Audio and Electroacoustics for June 1969, a special issue devoted to the fast Fourier transform.

Speech analysis-synthesis system based on homomorphic filtering.

Abstract: A digital speech analysis‐synthesis system based on a recently proposed approach to the deconvolution of speech is presented. The analyzer is based on a computation of the cepstrum considered as the inverse Fourier transform of the log magnitude of the Fourier transform. The transmitted parameters represent pitch and voiced unvoiced information and the low‐time portion of the cepstrum representing an approximation to the cepstrum of the vocal‐tract impulse response. In the synthesis, the low‐time cepstral information is transformed to an impulse response function, which is then convolved with a train of impulses during voiced portions or a noise waveform during unvoiced portions to reconstruct the speech. Since no phase information is retained in the analysis, phase must be regenerated during synthesis. Either a zero‐phase or minimum‐phase characteristic can be obtained by simple weighting of the cepstrum before transformation.

The finite Fourier transform

Abstract: The finite Fourier transform of a finite sequence is defined and its elementary properties are developed. The convolution and term-by-term product operations are defined and their equivalent operations in transform space are given. A discussion of the transforms of stretched and sampled functions leads to a sampling theorem for finite sequences. Finally, these results are used to give a simple derivation of the fast Fourier transform algorithm.

Directional energy spectra of the sea from photographs

Abstract: The image of the sun reflected by the sea surface is called sun glitter or sun glint. The relationship of the glitter pattern to the distribution of slopes on the surface is well known, but this effect is of little use in determining a wave spectrum because glitter does not provide a continuous representation of a wave parameter. A generalization of the glitter concept, in which the point image of the sun is replaced by a continuous skylight luminance function, is quite effective in providing a linear representation that can be recorded photographically. Then by the use of optical analysis it is possible to resolve the variations of density in the photographic emulsion into the components of the two-dimensional Fourier spectrum of the surface. When a transparency of a surface photograph is placed in one focal plane of a lens, the Fourier transform of the variations appears as light amplitude in the other focal plane. In the transform recording, density distribution represents slope distribution, wave direction is the azimuth angle, and wave number is proportional to the radius from the center. This information, in addition to the height and aspect angle of the camera, allows the energy spectrum of the surf are to be obtained.

Range-Doppler Imaging with Motion through Resolution Cells

Abstract: Doppler processing in pulsed radar is analyzed for time intervals which involve motion through range resolution cells, the emphasis being on the range-Doppler imaging of a rigid rotating body. The objective of the theory is to derive a method for compensating for motion through range and cross-range resolution cells. The compensation ion procedure described is compatible with optical data processing. With such a two-dimensional processor, the method permits simultaneous eous compensation for all points in the target field. The s consists of taking the Fourier transform in the range dimension, followed by a gentle distortion of this range-transform plane, and that followed by a two-dimensional Fourier transform. Two implementations with experimental results are briefly mentioned. One implementation is all optical and utilizes a holographic hyperbolic lens and/or holographic conical lens. The other implementation, involves applying the appropriate te distortion electronically as th " range sweeps" from the pulse train are received and put on film.

On the Fourier transform of the indicator function of a planar set.

Abstract: Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

On the response of beams to an arbitrary number of concentrated moving masses

Abstract: A theory describing the response of a beam under an arbitrary number of moving masses is developed. The theory is based on the Fourier technique and shows that, for a simply supported beam, the resonance frequency is lower with no corresponding decrease in maximum amplitude when the inertia is considered.

On the asymptotic behavior of the Fourier transform of the indicator function of a convex set

Abstract: Suppose C is a compact, convex subset of Rn, having a smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates (r= (x2 + + x2)12; =(x1/r, . . ., xn/r)) of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. Then it is known (cf. [1], [2]) that the function 'D(6) = supr rn + 1)/2 F(r, 0) is bounded on Sn1, provided AC is sufficiently smooth, and has everywhere positive Gaussian curvature. If AC has points of zero curvature, this need no longer be true (cf. [4]). The following, however, remains true.

Multiconductor transmission lines. Theory of natural modes and fourier integral applied to transient analysis

Abstract: In this paper, a theoretical formulation of a method of multiconductor transient analysis is developed. The method combines the use of the modified Fourier transform and the steady-state theory of natural modes. The virtues of this particular formulation are that the frequency dependence of parameters can be taken into account irrespective of the complexity of the expressions defining their steady-state values. The paper discusses the problem of the first pole to close when a power line is being energised from a source of infinite capacity. The argument is further extended to cover simultaneous pole closure of a circuit breaker. The mathematical representation of the source-side network is covered and two alternative methods of simulation of a multiconductor-source infeed are presented. Both methods are shown to retain the distributed-parameter nature of a source infeed, or a group of such infeeds, while achieving a reduction in the computation time that would otherwise be required to meet the needs of a fairly accurate form of representation. The numerical results of a series of digital-computer studies are presented. These illustrate various cases of practical important and highlight specific aspects of the behaviour of multiconductor systems.

Spectral Analysis of Detailed Vertical Wind Speed Profiles

Abstract: Cape Kennedy vertical wind speed profiles measured by spectral analysis using fast Fourier transform method

Representation and Analysis of Time‐Limited Signals Using a Complex Exponential Algorithm

Abstract: A complex exponential algorithm developed for the representation and analysis of time‐limited signals is defined, and its evaluation with respect to conventional discrete Fourier techniques is discussed. It is shown that for a given length of a signal containing discrete frequency information, sampled at least to the Nyquist criterion, the complex exponential algorithm can often provide increased frequency resolution over standard Fourier techniques. It is also shown that the complex exponential algorithm provides an improved mechanism over Fourier techniques for interpolation between points in a sampled signal containing discrete frequency components in the presence of broad‐band noise. The effects of noise in the complex exponential technique and the computational difficulties associated with the present complex exponential algorithm are also discussed.

An algorithm for the decomposition of a distribution into Gaussian components.

Abstract: An algorithm for the decomposition of a mixture of Gaussian components is given, partially in Algol-60 language, and experiences with its use on a digital computer are described. The algorithm contains three steps: (1) finding the mean of each component by the aid of a Fourier transform of the given density function and the method of decreasing the standard deviations (Medgyessi [1961]); (2) determining the standard deviations and frequencies of the components using the continued fraction approximation of the error function; and (3) testing results by the Kolmogorov-Smirnov method. Illustrative examples are given.

Image Deblurring and Aperture Synthesis Using a Posteriori Processing by Fourier-transform Holography

Abstract: Image deblurring and aperture synthesis using a posteriori processing by Fourier transform holographic spatial filtering

The correlation function of Gaussian noise passed through nonlinear devices

Abstract: This paper is concerned with the output autocorrelation function R^{y} of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes in R^{y} when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governing R^{y} is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devices y = x^{m} with m either an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculating R^{y} .

Dispersive Pulse Propagation Parallel to the Interfaces of a Laminated Composite

Abstract: This paper presents a theoretical analysis of the geometric dispersion of transient stress waves in a linearly elastic laminated composite. The loading is a uniform pressure of step-function time-dependence, applied to a half space. The laminates are perpendicular to the half-space boundary. The mathematical treatment is borrowed from the theory of wave propagation in rods. Fourier transforms are applied to time and the coordinate in the propagation direction. Inversion of the spatial transform by residues yields a formal solution in the form of an infinite series of integrals. Each of these integrals is the contribution to the transient response from a mode of sinusoidal wave propagation. Application of the saddle-point technique for long-time asymptotic approximation indicates that the low-frequency portion of the integral from the first mode gives the dominant contribution, called the head-of-the-pulse approximation. The form of the expression for the head-of-the-pulse approximation leads to the definition of a characteristic dispersion time τ. Since τ is a single quantity which describes the dispersion of the wave, it simplifies parametric studies. A closed-form algebraic expression for τ is presented, which has a simple dependence on the propagation distance and spacing of the laminates. Numerical examples for boron-epoxy and glass-epoxy laminates are given.

Diffraction of Partially Coherent Light by an Aberration-Free Annular Aperture

Abstract: Diffraction of quasimonochromatic partially space-coherent light by annular apertures, which are of fundamental importance when reflecting components are used, has been examined theoretically. The work on slit and circular apertures by previous workers has been briefly reviewed. Use has been made of Schell’s theorem which gives the relationship between the far-field intensity and the Fourier transform of the product of the source autocorrelation function and the correlation function across the diffracting aperture. Exponential, besinc, and sinc forms of the degree of coherence have been assumed and the results are presented for three typical values of the central obstruction of the circular aperture.

A short bibliography on the fast Fourier transform

Abstract: 166 L. E. Alsop and A. A. Nowroozi, “Fast Fourier analysis,” J. Geophys. Res., vol. 71, pp. 5482-5483, November 15, 1966. €3. Andrews, “A high-speed algorithm for the computer generation of Fourier transforms,” IEEE Trans. Computers (Short Notes), vol. C-17, pp. 373.375, April 1968. J. S . Bailey, “A fast Fourier transform without multiplications,” Proc. Symp. on Computer Processing in Communications, vol. 19, MKI Symposia Ser. New York: Polytechnic Press, 1969. V. Benignus, “Estimation of the coherence spectrum and its confidence interval using the fast Fourier transform,” this issue, pp. 145-150. G. D. Bergland, “The fast Fourier transform recursive equations for arbitrary length records,” Math. Computation, vol. 21, pp, 236-238, April 1967. -9 “A fast Fourier transform algorithm using base eight iterations,” Math. Computation, vol. 22, pp. 275-279, April 1968. -, “A fast Fourier transform algorithm for realvalued series,” Commun. A C M , vol. 11, pp. 703--710, October 1968. -, “A radix-eight fast Fourier transform subroutine for real-valued series,” this issue, pp. 138144. -, “A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published). “Fast Fourier transform hardware implementations. I. An overview. 11. A survey,’’ this issue,

Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

Abstract: The theory of generalized functions and Fourier transforms is used to derive the Laplace‐type expansion for r12nYlm(θ12, φ12). This approach leads naturally to a general formula for the Dirac delta‐function terms which occur when n≤−3 and n − l is odd.

Die Systemtheorie der homogenen Schichten

Abstract: A systems theory describing signal transmission in neuronal layer structures is presented. Such structures having the quality of linearity and homogenity can easily be treated by using a multiple Fourier transformation method with respect to the space coordinates and to the time. The characteristic laws of this transformation are discussed. In analogy to linear networks the transfer function, the impulse response function and the step response function are defined which characterizes the layer system. Appropriate test functions are derived. Typical structures of standing and moving patterns are finally discussed. Biological application and simulation of the theory by a system using coherent optics will be presented later.

Analytical Evaluation of Multicenter Integrals of r12−1 with Slater‐Type Atomic Orbitals. V. Four‐Center Integrals by Fourier‐Transform Method

Abstract: The four‐center integral of r12−1 with Slater‐type atomic orbitals is evaluated analytically The Fourier‐transform convolution theorem is used to express the integral as an infinite sum in which the internuclear angles appear in spherical harmonics, and the internuclear distances in integrals over spherical Bessel functions and exponential‐type integrals These “radial” integrals are evaluated as convergent infinite expansions by contour integration techniques The formulas are valid for general values of the n, l, m, ζ parameters of the orbitals and for general nonzero values of the internuclear distance vectors

Digital Beamforming in the Frequency Domain

Abstract: A recent proposal to use digital Fourier transformation in forming acoustic beams is discussed tutorially. Relations to other procedures for beamforming, and for spectral analysis, are given. Need is shown for a lower limit on the data batch size.

Fourier transform recording with random phase shifting

Abstract: In making a record of the exact Fourier transform of an array of beams of electromagnetic radiation, the phase of each of a substantial fraction of the beams is shifted by a constant amount before recording the transform.