Showing papers on "Hartley transform published in 1977"
TL;DR: A new algorithm for evaluating Hankel (Fourier–Bessel) transforms numerically with enhanced speed, accuracy, and efficiency is outlined.
Abstract: We outline here a new algorithm for evaluating Hankel (Fourier–Bessel) transforms numerically with enhanced speed, accuracy, and efficiency. A nonlinear change of variables is used to convert the one-sided Hankel transform integral into a two-sided cross-correlation integral. This correlation integral is then evaluated on a discrete sampled basis using fast Fourier transforms. The new algorithm offers advantages in speed and substantial advantages in storage requirements over conventional methods for evaluating Hankel transforms with large numbers of points.
293 citations
TL;DR: In this article, the authors give a criterion for the intersection of two projections in Hilbert space to be a projection of finite-dimensional range, which is applied to Schrodinger operators in L2(Rn) and to the problem of determining whether there are functions f and its Fourier transform having prescribed support.
131 citations
TL;DR: The Walsh-Hadamard transform has recently received increasing attention in engineering applications due to the simplicity of its implementation and to its properties which are similar to the familiar Fourier transform.
Abstract: The Walsh-Hadamard transform has recently received increasing attention in engineering applications due to the simplicity of its implementation and to its properties which are similar to the familiar Fourier transform. The transform matrices found so far to possess fast algorithms are the naturally ordered and dyadically ordered matrices, whose algorithms are similar to the Cooley-Tukey algorithm, and to the machine-oriented algorithm of Corinthios [2], respectively.
65 citations
01 Feb 1977
TL;DR: Transforms using number theoretic concepts developed as a method for fast and error-free calculation of finite digital convolution are shown to be ideally suited to digital computation by taking into account quantization of amplitude as well as time in their definition.
Abstract: A correct proof of Theorem 2 from the paper "Number Theoretic Transforms to Implement Fast Digital Convolutions," giving necessary and sufficient conditions for the modular Fourier transform, is presented. A counterexample to Theorem 1 of the above paper is also given.
46 citations
TL;DR: The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT and is used to filter a two-dimensional picture, and the results are presented with a comparison to the standard FFT.
Abstract: A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.
44 citations
IBM1
TL;DR: These transforms, which under certain conditions can be computed via fast transform algorithms allow the implementation of digital filters with better efficiency and accuracy than the fast Fourier transform (FFT).
Abstract: In this paper pseudo Fermat number transforms (FNT's) are discussed. These transforms are defined in a ring of integers modulo an integer submultiple of a pseudo Fermat number, and can be computed without multiplications while allowing a great flexibility in word length selection. Complex pseudo FNT's are then introduced and are shown to relieve some of the length limitations of conventional Fermat number transforms (FNT's). These transforms, which under certain conditions can be computed via fast transform algorithms allow the implementation of digital filters with better efficiency and accuracy than the fast Fourier transform (FFT).
44 citations
01 Jul 1977
TL;DR: In this article, a 1D algorithm using the Hankel transform of the section of the function is described, which can avoid the use of the 2D FFT algorithm due to the loss of symmetry due to sampling and to a waste in storage requirements.
Abstract: Computing the Fourier transform of a circularly symmetric function is often necessary in optics. Use of the 2-D FFT algorithm leads to loss of the symmetry because of the sampling and to a waste in storage requirements; to avoid these inconveniences, a 1-D algorithm is described using the Hankel transform of the section of the function.
20 citations
TL;DR: It is indicated that this approach to Fourier transform processing of multicomponent exponential functions yields processed outputs whose peaks are proportional to Ai directly and offers an advantageously different noise performance.
Abstract: Conventional Fourier transform processing of multicomponent exponential functions of the form f(t) = ?iAie-?it commences by forming the product exf(ex). This note is concerned with an alternative starting point?the formation of the x-derivative, f'(ex). It is indicated that this approach: 1) yields processed outputs whose peaks are proportional to Ai directly; 2) offers an advantageously different noise performance; 3) can deal with functions containing a constant (D. C.) bias level; and 4) requires in practice only the formation of first differences.
19 citations
TL;DR: The fast Fourier transform andd the fast Walsh transform are too slow for some real-time applications, so these are replaced by simple, modular logic solutions.
Abstract: The fast Fourier transform andd the fast Walsh transform are too slow for some real-time applications. For binary data, an 'instant' Fourier transform is based on harmonic analysis in a space of 2 n -tuples of 0s and 1s. Simple, modular logic finishes transforming 2n real-time serial binary data one clock pulse after the last datum arrives.
17 citations
01 Dec 1977
15 citations
TL;DR: It is found that the rewriting of the Fourier transform to scale data only when arithmetic overflow occurs, rather than before each pass, results in a twofold increase in the available dynamic range.
TL;DR: In this paper, the orthogonality conditions that must be fulfilled by the transform factor α of a NTFT of length N, are proven based upon the possibility of cancelling all nonzero factors of the form (αq-1), q = 1, 2,..., N - 1.
Abstract: The proof of the orthogonality conditions that must be fulfilled by the transform factor α of a NTFT of length N, is based upon the possibility of cancelling all nonzero factors of the form (αq- 1), q = 1, 2,..., N - 1. In a residue ring containing zero divisors, this is not allowed, unless all such factors can be shown not to be divisors of zero. It is shown that this is the case, when a is any primitive Nth root of unity, N being an allowed transform legnth. At the same time, a property is established that helps to reduce the amount of searching needed to find suitable transform factors.
IBM1
TL;DR: The sinc and cosinc transform (SCT) as mentioned in this paper uses Walsh functions to obtain the Fourier transform, which converts a staircase approximation of a function to a set of sinc terms in the frequency domain.
Abstract: A new transform, the sinc and cosinc transform, uses Walsh functions to obtain the Fourier transform. This technique converts a staircase approximation of a function to a set of sinc and cosinc terms in the frequency domain that is equivalent to the Fourier transform. The calculation is slower than the fast Fourier transform (FFT) but is devoid of aliasing. The interpolation and scaling in the frequency domain are built in, and any frequency point may be chosen without changing the number or spacing of the samples in the time domain. The intervening set of coefficients is computed more rapidly than those obtained using the fast Hadamard transform.
TL;DR: In this paper, the differential transform is used to represent some point sources in operational form, where the sources can also be represented by integrals or integral transforms, but in order to solve boundary value problems, it is simpler to use the operational representation than the integral or integral transform representation.
TL;DR: In this paper, a numerical method of holographic reconstruction of 2D fields is presented, where the 2D Fourier transform of the field is generated by using the slowness curve of the medium.
Abstract: A numerical method of holographic reconstruction of 2-dimensional fields is presented. Reconstructions are obtained indirectly, by generating the 2-dimensional Fourier transform of the field by using the slowness curve of the medium. Wideband data can be handled efficiently by this method, which also works in dispersive and anisotropic space. Examples of monochromatic and multifrequency holographic reconstructions are shown.
TL;DR: In this paper, a modified variant of the successive convolution unfolding method is proposed by considering the convolution product only over a finite size range where the distribution could be non-zero.
Abstract: A diffraction profile can be recorded for only a finite range around the peak. This leads to spurious oscillations in the block-size distribution determined by means of the Fourier transform. The Bertaut [Acta Cryst. (1952), 5, 117-121] correction gives an approximate distribution g1(m) related to the real one g(m) by a convolution product with a function d(m), of the type (sin x)/x. A modified variant of the successive-convolution unfolding method is proposed by considering the convolution product only over a finite size range where the distribution could be non-zero. The method was tested for two hypothetical distributions. A procedure to set the limits of the size interval is suggested.
TL;DR: In this article, it was shown that the diffraction integral transform involves, in special cases, the Fresnel transform and the Rayleigh transform in diffraction theory, the Gauss transform in diffusion theory, and the Lorentz transform in spectrum analysis of light emitted from thermally moving molecules.
Abstract: It is shown that the diffraction integral transform involves, in special cases, the Fresnel transform and the Rayleigh transform in the diffraction theory, the Gauss transform in the diffusion theory, the Lorentz transform in the spectrum analysis of light emitted from thermally moving molecules, and the modified Hilbert transform. The sampling theorem in the diffraction integral transform is proved by means of the functional analytic method, and the interpolation functions are shown in respective cases. This theorem has applications to fast transformations by the digital computer.