Showing papers on "Discrete-time Fourier transform published in 1969"
IBM1
TL;DR: A description of the alogorithm and its programming is given here and followed by a theorem relating its operands, the finite sample sequences, to the continuous functions they often are intended to approximate.
Abstract: The advent of the fast Fourier transform method has greatly extended our ability to implement Fourier methods on digital computers A description of the alogorithm and its programming is given here and followed by a theorem relating its operands, the finite sample sequences, to the continuous functions they often are intended to approximate An analysis of the error due to discrete sampling over finite ranges is given in terms of aliasing Procedures for computing Fourier integrals, convolutions and lagged products are outlined
833 citations
TL;DR: A computational algorithm for numerically evaluating the z -transform of a sequence of N samples is discussed, based on the fact that the values of the z-transform on a circular or spiral contour can be expressed as a discrete convolution.
Abstract: A computational algorithm for numerically evaluating the z -transform of a sequence of N samples is discussed. This algorithm has been named the chirp z -transform (CZT) algorithm. Using the CZT algorithm one can efficiently evaluate the z -transform at M points in the z -plane which lie on circular or spiral contours beginning at any arbitrary point in the z -plane. The angular spacing of the points is an arbitrary constant, and M and N are arbitrary integers. The algorithm is based on the fact that the values of the z -transform on a circular or spiral contour can be expressed as a discrete convolution. Thus one can use well-known high-speed convolution techniques to evaluate the transform efficiently. For M and N moderately large, the computation time is roughly proportional to (N+M) \log_{2}(N+M) as opposed to being proportional to N . M for direct evaluation of the z -transform at M points.
608 citations
TL;DR: In this paper, two schemes for numerical solution of the Navier-Stokes equations at moderate Reynolds number are discussed, and the essential ingredient of the present methods is the use of the fast Fourier transform.
Abstract: Two schemes for the numerical solution of the Navier‐Stokes equations at moderate Reynolds number are discussed. The essential ingredient of the present methods is the use of the fast Fourier transform. In one scheme, discrete Fourier transformation is used to compute the convolution sums appearing in the formally Fourier‐transformed Navier‐Stokes equations. This results in an aliasing‐free, energetically conservative (when ν = 0) scheme that can be used convincingly as a model on which to test turbulence theories. The second scheme calculates in real space. Fast Fourier transform is used to solve Poisson's equation for the pressure. The latter scheme offers the advantages of speed and flexibility. The schemes are critically compared and a survey of applications is made.
214 citations
IBM1
TL;DR: In this paper, the finite Fourier transform of a finite sequence is defined and its elementary properties are developed, and the convolution and term-by-term product operations are defined and their equivalent operations in transform space.
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.
165 citations
IBM1
TL;DR: In this article, an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm is presented, which leads to approximate upper and lower bounds on the root-mean-square error.
Abstract: This paper contains an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simulated fixed-point machine and their comparison with the error upper bound.
164 citations
01 Jan 1969
TL;DR: This chapter contains sections titled: Introduction, An Algorithm Suggested by Chirp Filtering, and An Algorithm Suggested By ChirP Filtering.
Abstract: This chapter contains sections titled: Introduction, An Algorithm Suggested by Chirp Filtering
121 citations
TL;DR: Fast Fourier analysis (FFA) and fast Fourier synthesis (FFS) algorithms are developed for computing the discrete Fourier transform of a real series, and for synthesizing a realseries from its complex Fourier coefficients.
Abstract: Fast Fourier analysis (FFA) and fast Fourier synthesis (FFS) algorithms are developed for computing the discrete Fourier transform of a real series, and for synthesizing a real series from its complex Fourier coefficients. A FORTRAN program implementing both algorithms is given in the Appendix.
99 citations
13 Oct 1969
TL;DR: A data transmission system in which the transmitted signal is the Fourier transform of the original data sequence and the demodulator is a discrete Fourier transformer, and it is shown, via computer simulation and computation of the variances of errors, how the system corrects linear channel distortion.
Abstract: The development of rapid algorithms for computation of the discrete Fourier transform has encouraged the use of this transform in the design of communication systems. Here we describe and analyze a data transmission system in which the transmitted signal is the Fourier transform of the original data sequence and the demodulator is a discrete Fourier transformer. This system is a realization of the frequency division multiplexing strategy known as “parallel data transmission”, and it is constructed in this manner so that the data demodulator, after analog to digital conversion, may be a computer program employing one of the fast Fourier transform algorithms. The system appears attractive in that it may be entirely implemented by digital circuitry. We study the performance of this system in the presence of typical linear channel characteristics. It is shown, via computer simulation and computation of the variances of errors, how the system corrects linear channel distortion.
59 citations
TL;DR: Two methods for FFT of one-dimensional arrays of data to be fast Fourier transformed are presented-one efficient when data storage is only slightly larger than available internal memory, and one when data is much larger.
Abstract: Occasionally, arrays of data to be fast Fourier transformed (FFT'ed) are too large to fit in internal computer memory, and must be kept on an external storage device. This situation is especially serious for one-dimensional arrays, since they cannot be factored along the natural cleavage planes, as multi-dimensional arrays can. Two methods for FFT of such data are presented-one efficient when data storage is only slightly larger than available internal memory, and one when data is much larger. A FORTRAN program based on these methods is available.
56 citations
TL;DR: The fast Fourier transform is considered to owe its speed to the fact that a certain matrix, none of whose elements is zero, can be factored into matrices with very many zeros as mentioned in this paper.
Abstract: The fast Fourier transform is considered to owe its speed to the fact that a certain matrix, none of whose elements is zero, can be factored into matrices with very many zeros. This paper describes and discusses a procedure for explicitly carrying out such a factorization.
TL;DR: A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published).
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,
Patent•
22 Oct 1969TL;DR: 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 as discussed by the authors.
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.
TL;DR: In this article, the application of Fourier transform methods in elasticity problems is discussed and two different, possible methods of approach and their limitations for the solution of the above problem are presented.
Abstract: The application of Fourier transform methods in elasticity problems is discussed. An example of a half-space with the external load extending to infinity is chosen to illustrate the problem that Fourier transforms cannot be obtained in a strict mathematical sense. Two different, possible methods of approach and their limitations for the solution of the above problem are presented.
TL;DR: An efficient method of computing spectrum and cross-spectrum of large scale aero-magnetic field (or of any other two-dimensional field) has been developed and programmed for a digital computer and reduces greatly computational time and storage requirements.
Abstract: An efficient method of computing spectrum and cross-spectrum of large scale aero-magnetic field (or of any other two-dimensional field) has been developed and programmed for a digital computer. The method uses fast Fourier transform techniques. Briefly, the method is as follows: a digitized aeromagnetic map is divided into a number of rectangular blocks. Fourier transforms of these blocks are computed using a two-dimensional fast Fourier transform method. Finally, the amplitude of the Fourier transforms is averaged to give the desired spectrum. Computation of cross-spectrum follows the same lines. In fact, the same programme may be used to a compute the spectrum as well as cross-spectrum. The method has a number of computational advantages, in particular it reduces greatly computational time and storage requirements. The programme has been tested on synthetic data as well as on real aeromagnetic data. It took less than 30 seconds on an IBM 360/50 computer to compute the spectrum of an aeromagnetic map covering an area of approximately 4500 square miles.
TL;DR: The following method, which relies only on arithmetical operations available in all programming languages, is used for generating functions in the form of polynomials and infinite power series.
Abstract: MUCH use is made in combinatorial problems of generating functions in the form of polynomials and infinite power series, these being obtained by the manipulation of other algebraic expressions. In order to save time and improve accuracy in the evaluation of the coefficients, one can, of course, make use of computer programs for doing algebra1,2. But it is often easier to use the following method which relies only on arithmetical operations available in all programming languages.
Patent•
03 Nov 1969TL;DR: A fast Fourier transform processor and associated process where an input sequence of samples is broadcast to each of a plurality of parallel processing elements where sets of accumulated sums of products of these samples with appropriate trigonometric function values are maintained is described in this paper.
Abstract: A fast Fourier transform processor and associated process wherein an input sequence of samples is broadcast to each of a plurality of parallel processing elements where sets of accumulated sums of products of these samples with appropriate trigonometric function values are maintained. These sets of accumulated sums are then individually Fourier transformed in parallel to form the Fourier coefficients corresponding to the original input sequence.
TL;DR: In this paper, it is shown how to determine any Fourier coefficient (spatial frequency) in a real two-dimensional distribution of illumination by allowing the distribution to throw the shadow of a suitably placed grid onto an observation plane, where the contrast of the shadow measures the modulus of the Fourier coefficients and its position measures the phase of the coefficient.
Abstract: It is possible to determine any Fourier coefficient (spatial frequency) in a real two-dimensional distribution of illumination by allowing the distribution to throw the shadow of a suitably placed grid onto an observation plane, where the contrast of the shadow measures the modulus of the Fourier coefficient and its position measures the phase of the coefficient. A pilot model of a scheme for reading the Arabic numerals 0, 1, …, 9 using only three Fourier coefficients has been set up and run successfully. The possibilities of the method are assessed.
TL;DR: In this article, the operation of convolution is explored-starting with discrete rather than continuous convolution because of the relative ease of comprehension involved and a proof of the convolution theorem will show that convolution and transform analysis are closely related.
Abstract: Few mathematical operations are more important to the engineer than convolution and transform analysis. In this article, the operation of convolution is explored?starting with discrete rather than continuous convolution because of the relative ease of comprehension involved. With this foundation, the study is extended to continuous convolution. A proof of the convolution theorem will show that convolution and transform analysis are closely related. Of much more interest, however, is an intuitive explanation of why convolution and transform analysis techniques lead to exactly the same solution of a given problem. Perhaps the two most important applications of convolution deal with the analysis of linear systems and the sums of independent random variables?the latter problem being used to introduce discrete convolution.
01 Jan 1969
TL;DR: Discrete Fourier transform method for factoring spectral density functions, calculating absolute error as discussed by the authors, is used to calculate absolute error of spectral density function, which is a function of spectral distribution.
Abstract: Discrete Fourier transform method for factoring spectral density functions, calculating absolute error
01 Nov 1969
TL;DR: This is an introduction to the basic elements of Fourier analysis of a time series that is a sum of deterministic components and a stationary random process.
Abstract: : This is an introduction to the basic elements of Fourier analysis of a time series that is a sum of deterministic components and a stationary random process. The various mathematical concepts relating to Fourier analysis are presented in a basically intuitive manner. The intended audience consists of economists and data processors. (Author)
TL;DR: The class of digital filters which have an impulse response of finite duration and are implemented by means of circular convolutions performed using the discrete Fourier transform is considered and a least upper bound is obtained for the maximum possible output of a circular convolution for the general case of complex input sequences.
Abstract: When implementing a digital filter, it is important to utilize in the design a bound or estimate of the largest output value which will be obtained. Such a bound is particularly useful when fixed point arithmetic is to be used since it assists in determining register lengths necessary to prevent overflow. In this paper we consider the class of digital filters which have an impulse response of finite duration and are implemented by means of circular convolutions performed using the discrete Fourier transform. A least upper bound is obtained for the maximum possible output of a circular convolution for the general case of complex input sequences. For the case of real input sequences, a lower bound on the least upper bound is obtained. The use of these results in the implementation of this class of digital filters is discussed.
TL;DR: A modified fast Fourier transform is used in a hybrid computer program to permit processing of tracking data during a run to yield the human operator's describing function almost immediately after the data-taking period.
Abstract: A modified fast Fourier transform (FFT) is used in a hybrid computer program to permit processing of tracking data during a run to yield the human operator's describing function almost immediately after the data-taking period. The computer processing time is substantially reduced at no cost in accuracy.
TL;DR: In this article, certain striking similarities between the discrete characterisation of continuous signals by periodic sampling and by the Poisson transform are revealed, and they are used to compare the two methods.
Abstract: Certain striking similarities between the discrete characterisation of continuous signals by periodic sampling and by the Poisson transform are revealed here.