Showing papers on "Discrete-time Fourier transform published in 1977"
TL;DR: In this article, a theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions, which are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.
Abstract: A theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions. The methods discussed here are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.
899 citations
TL;DR: Two recently developed ideas, the conversion of a discrete Fourier transform to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms.
Abstract: Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented
331 citations
IBM1
TL;DR: It is shown how the Chinese Remainder Theorem can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions and can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm.
Abstract: It is shown how the Chinese Remainder Theorem (CRT) can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions. Then, special algorithms are developed which, compute the relatively short convolutions in each of the dimensions. The original suggestion for this procedure was made in order to extend the lengths of the convolutions which one can compute with number-theoretic transforms. However, it is shown that the method can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm. Some of the short convolutions are computed by methods in an earlier paper by Agarwal and Burrus. Recent work of Winograd, consisting of theorems giving the minimum possible numbers of multiplications and methods for achieving them, are applied to these short convolutions.
257 citations
IBM1
TL;DR: A new approach to the computation of the discrete Fourier transform (DFT) with significantly reduced number of multiplication operations; it does not increase the number of addition operations in many cases.
Abstract: Recently, Dr. Shmuel Winograd discovered a new approach to the computation of the discrete Fourier transform (DFT). Relative to fast Fourier transform (FFT), the Winograd Fourier transform algorithm (WFTA) significantly reduces the number of multiplication operations; it does not increase the number of addition operations in many cases. This paper introduces the new algorithm and discusses the operations comparison problem. A guide for programming is included, as are some preliminary running times.
178 citations
TL;DR: The general conditions for these mappings to be unique and cyclic are given, and the application to discrete Fourier transform (DFT) and convolution evaluation is considered.
Abstract: The mapping of one-dimensional arrays into two- or higher dimensional arrays is the basis of the fast Fourier transform (FFT) algorithms and certain fast convolution schemes. This paper gives the general conditions for these mappings to be unique and cyclic, and then considers the application to discrete Fourier transform (DFT) and convolution evaluation.
172 citations
TL;DR: In this article, a method for calculating structure factors by Fourier inversion of a model electron density map is presented, which is 3½ to 7 times less expensive than conventional methods for non-centrosymmetric space groups.
Abstract: A method is presented for calculating structure factors by Fourier inversion of a model electron density map. The cost of this method and of the standard methods are analyzed as a function of number of atoms, resolution, and complexity of space group. The cost functions were scaled together by timing both methods on the same problem, with the same computer. The FFT method is 3½ to 7 times less expensive than conventional methods for non-centrosymmetric space groups.
105 citations
102 citations
TL;DR: A generalized mathematical theory of holor algebra is used to manipulate coefficient arrays needed to generate computational equations which involve elements from throughout the two-dimensional array rather than operating on individual rows and columns.
Abstract: A mathematical development is presented for a direct computation of a two-dimensional fast Fourier transform (FFT). A generalized mathematical theory of holor algebra is used to manipulate coefficient arrays needed to generate computational equations. The result is a set of equations which involve elements from throughout the two-dimensional array rather than operating on individual rows and columns. Preliminary digital computer calculations verify the accuracy of the technique and demonstrate a modest saving of computation time as well.
74 citations
Patent•
IBM1
TL;DR: In this paper, a method for determining the initial values of the coefficients of a transversal equalizer in a data transmission system in which the transmission channel creates frequency shift is presented.
Abstract: A method of and apparatus for determining during an initial training period the initial values of the coefficients of a transversal equalizer in a data transmission system in which the transmission channel creates frequency shift. The received periodic training sequence is modulated by a time-domain window signal whose Fourier transform exhibits a relatively flat central peak and has comparatively low values in the vicinity of those frequencies which are a multiple of the inverse of the period of the transmitted sequence, and the discrete Fourier transform Wk of the modulated signal is computed. The values of the coefficients of the equalizer are obtained by computing the inverse discrete Fourier transform of the ratio Fk =Zk /Wk, where Zk is the discrete Fourier transform of the transmitted sequence.
58 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.
TL;DR: Two methods of computing the complex cepstrum of a two-dimensional (2-D) signal are explored and the relationship among the nonzero regions of a signal, its inverse, and its cEPstrum is explored.
Abstract: In this paper we shall explore two methods of computing the complex cepstrum of a two-dimensional (2-D) signal. The two principal methods for computing 1-D cepstra, using discrete Fourier transforms (DFT's) and the complex logarithm function or using a recursion relation for minimum-phase signals, may be extended to two dimensions. These two algorithms are developed and simple examples of their use are given. As a matter of course, we shall also be drawn into considering the definitions of 2-D causality and 2-D minimum-phase signals. In addition, we shall explore the relationship among the nonzero regions of a signal, its inverse, and its cepstrum.
TL;DR: For an elastic material with an infinite circular cylindrical hole, the exact solution due to a pressure on a finite length of the cylinder is obtained as a function of the Laplace transform parameters on time and Fourier transform parameter on the z-coordinate as discussed by the authors.
Abstract: For an elastic material with an infinite circular cylindrical hole, the exact solution due to a pressure on a finite length of the cylinder is obtained as a function of the Laplace transform parameter on time and Fourier transform parameter on the z-coordinate (the axis of the cylinder). The applied pressure is a function of the time and the position z. Numerical inversion of the Laplace and Fourier transforms are required to determine the field quantities in the time and space parameters. In the far field, the inverse Fourier transform can be obtained by an asymptotic expansion. It remains to obtain the inverse Laplace transform numerically. We have found that for cylinders whose radius is small compared with the smallest wavelength of interest, an analytical solution can be obtained. Graphical results for the cases of instantaneous explosion and progression of the detonation with constant velocity are given. In both cases an exponential decay of the explosion pressure is assumed.
TL;DR: In this article, the Fourier transform of the gravity field due to a finite dipping dike is derived and its real and imaginary components are separated and simple relations that can be used to estimate the unknown parameters of the dike.
Abstract: The Fourier transform of the gravity field due to a finite dipping dike is derived and its real and imaginary components are separated. Analysis of these two functions in a certain high-frequency range yields simple relations that can be used to estimate the unknown parameters of the dike. The theoretical considerations are tested on synthetic data after performing the discrete Fourier transform (DFT), and the validity of the method of interpretation is established from a comparison of the actual and estimated parameters.
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.
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.
TL;DR: In this article, a discrete filtering technique based on circular convolution is presented, which is shown to compare favorably with DFT and FFT filtering, in terms of error accumulation.
Abstract: A discrete filtering technique based on circular convolution is presented. The discrete Hilbert transform (DHT), in matrix form and other matrices for filtering by circular convolution, is shown to compare favorably with DFT and FFT filtering. The comparison is presented in terms of error accumulation.
TL;DR: In this article, a simple computational procedure is discussed by which it is possible to correct fourier NMR spectra automatically for all amplitude and phase distortions resulting from signal-conditioning circuits and from anomalies in quadrature phase detection.
TL;DR: In this paper, a simple Fourier series algorithm is presented for the determination of the phase, and hence the other optical constants from measurements of the reflectance, and the advantages of the proposed procedure and its mathematical equivalence with the Kramers-Kronig relations are discussed.
Abstract: A simple Fourier series algorithm is presented for the determination of the phase, and hence the other optical constants from measurements of the reflectance. The advantages of the proposed procedure and its mathematical equivalence with the Kramers-Kronig relations are discussed.
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.
TL;DR: In this paper, the gravity effect of an infinite horizontal trapezoidal prism is derived and its Fourier spectrum is analyzed so as to yield information about four parameters of the causative structure, namely the depths to the upper and lower surfaces, width of the upper surface, and the inclination of the sides.
Abstract: The gravity effect of an infinite horizontal trapezoidal prism is derived and its Fourier spectrum is analyzed so as to yield information about four parameters of the causative structure, namely the depths to the upper and lower surfaces, width of the upper surface, and the inclination of the sides. In order to test the applicability of the method, synthetic data are constructed by digitizing the theoretical gravity effect. Subsequently, the corresponding Discrete Fourier Transform (DFT) is obtained. The parameters evaluated from the DFT are observed to be sufficiently close to the chosen values.
TL;DR: In this paper, Chandra et al. consider lie groups with complex Lie algebras lc and fc and show that a unique germ of a C w function e° at s0 := K in S := L/K such that e°(s0) = 1 and xe° = o(x)e° (x in b) is a spherical function.
Abstract: 1. Let L D K be lie groups with complex Lie algebras lc and fc. Assume tc has a linear complement b in \c which is a subalgebra. For any o in LieHomc(b, C) there is a unique germ of a C w function e° at s0 := K in S := L/K such that e°(s0) = 1 and xe° = o(x)e° (x in b). Now suppose S is connected, K is compact, and e extends to an element of C°*(S). Then (HarishChandra)
TL;DR: Observed patterns from the Beatquency Domain suggest usefulness of this method in sleep cycle detection and a new and unusual application of the FFT on heart rate data.
Abstract: Fourier analysis has proven to be a vital mathematical tool in many areas of research, but rapid methods for calculating frequency content of sampled data using discrete Fourier transform (FFT) require periodic sampling. Unfortunately, beat-by-beat heart rate is an aperiodic series of events in the time domain. This report describes a new and unusual application of the FFT on heart rate data. The beat-by-beat intervals are represented as the magnitude of a periodically sampled function. When the Fourier transform is applied to these data, we obtain pseudofrequency information from what we call the Beat-quency Domain. Observed patterns from the Beatquency Domain suggest usefulness of this method in sleep cycle detection.
01 May 1977
TL;DR: This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process, which is extremely valuable on small disc based computers.
Abstract: Conventional two dimensional fast Fourier transforms become very slow if the size of the matrix becomes too large to be contained in memory. This is due to the transposition of the matrix that is required. This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process. This algorithm is extremely valuable on small disc based computers.
TL;DR: An interpolation technique (interpolation by repetitive convolution) is proposed which yields values accurate enough for plotting purposes and which lie within the limits of calibration accuracies and is shown to operate faster than zero fill, since fewer operations are required.
Abstract: Zero fill, or augmentation by zeros, is a method used in conjunction with fast Fourier transforms to obtain spectral spacing at intervals closer than obtainable from the original input data set. In the present paper, an interpolation technique (interpolation by repetitive convolution) is proposed which yields values accurate enough for plotting purposes and which lie within the limits of calibration accuracies. The technique is shown to operate faster than zero fill, since fewer operations are required. The major advantages of interpolation by repetitive convolution are that efficient use of memory is possible (thus avoiding the difficulties encountered in decimation in time FFTs) and that is is easy to implement.
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.
IBM1
TL;DR: One "General-N" (i.e. many allowable DFT sizes (N) but certainly not any vector size) complex WFTA programming technique is described.
Abstract: The Winograd Fourier Transform Algorithm (WFTA) requires about 20% of the multiplications used in an optimized FFT, while the number of additions remains unchanged. This paper describes one "General-N" (i.e. many allowable DFT sizes (N) but certainly not any vector size) complex WFTA programming technique.
TL;DR: In this paper, the application of fast Fourier transformation techniques to the analysis of experimental X-ray and neutron diffraction patterns from amorphous materials is discussed and compared with conventional techniques using Filon's quadrature.
TL;DR: A classification of methods for generating discrete Fourier transform pairs is given, followed by a table of 29 pairs that shows hundreds of additional nonobvious finite identities can be deduced by using the Rayleigh-Parseval formula and convolutions.
Abstract: A classification of methods for generating discrete Fourier transform pairs is given, followed by a table of 29 pairs. Many of these are new, whereas some have been collected from various literature sources. We have tried to make the table interesting rather than comprehensive. The generalization of the Gaussian sums is a good example. Hundreds of additional nonobvious finite identities can be deduced by using the Rayleigh-Parseval formula and convolutions.
01 Jan 1977