Showing papers on "Fractional 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
01 Jan 1969
TL;DR: A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed, which provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.
Abstract: The introduction of the fast Fourier transform algorithm has led to the development of the Fourier transform image coding technique whereby the two-dimensional Fourier transform of an image is transmitted over a channel rather than the image itself. This devlopement has further led to a related image coding technique in which an image is transformed by a Hadamard matrix operator. The Hadamard matrix is a square array of plus and minus ones whose rows and columns are orthogonal to one another. A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed. Since only real number additions and subtractions are required with the Hadamard transform, an order of magnitude speed advantage is possible compared to the complex number Fourier transform. Transmitting the Hadamard transform of an image rather than the spatial representation of the image provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.
634 citations
TL;DR: This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey, and includes an efficient method for permuting the results in place.
Abstract: This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension n of the transform is factored (if possible), and n/p elementary transforms of dimension p are computed for each factor p of n . An improved method of computing a transform step corresponding to an odd factor of n is given; with this method, the number of complex multiplications for an elementary transform of dimension p is reduced from (p-1)^{2} to (p-1)^{2}/4 for odd p . The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine.
534 citations
TL;DR: An efficient Walsh transform computation algorithm is derived which is analogous to the Cooley-Tukey algorithm for the complex-exponential Fourier transform.
Abstract: The discrete, orthogonal Walsh functions can be generated by a multiplicative iteration equation. Using this iteration equation, an efficient Walsh transform computation algorithm is derived which is analogous to the Cooley-Tukey algorithm for the complex-exponential Fourier transform.
172 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
TL;DR: In this paper, the problem of the Hilbert transform of a sequence of discrete sample values is analyzed, and the direct Hilbert transform and the indirect Hilbert transform through intermediate Fourier transformation are compared; the latter is found to be more efficient with respect to computing time.
83 citations
01 Jan 1969
TL;DR: In this paper, the Hilbert transform relations, as they apply to sequences and their z-transforms, and also as the number of data samples taken in the Discrete Fourier Transforms becomes infinite, are discussed.
Abstract: The Hilbert transform has traditionally played an important part in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform plays a similar role in digital signal processing. In this paper, the Hilbert transform relations, as they apply to sequences and their z-transforms, and also as they apply to sequences and their Discrete Fourier Transforms, will be discussed. These relations are identical only in the limit as the number of data samples taken in the Discrete Fourier Transforms becomes infinite. The implementation of the Hilbert transform operation as applied to sequences usually takes the form of digital linear networks with constant coefficients, either recursive or non-recursive, which approximate an all-pass network with 90° phase shift, or two-output digital networks which have a 90° phase difference over a wide range of frequencies. Means of implementing such phase shifting and phase splitting networks are presented.
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.
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,
TL;DR: In this paper, the theory of generalized functions and Fourier transforms is used to derive the Laplace type expansion for r12nYlm(θ12, φ12).
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.
TL;DR: The transform presented in this paper applies to functions which describe logic network behavior, and both form and development of this transform pair resembles the Fourier transform in harmonic analysis.
Abstract: The transform presented in this paper applies to functions which describe logic network behavior. Given a function G defined over a finite domain, it is shown that G(u) = Et F(t)ut for each element u in the domain, where finite-field arithmetic is assumed. Here, function F is the transform of G, and it is shown that F(t) = Eu G(u)(-u)-t for each integer t in a finite set. Both form and development of this transform pair resembles the Fourier transform in harmonic analysis.
TL;DR: In this article, Walsh functions are used in transform Spectroscopy to replace the sinusoidal functions appearing in the Fourier transform, and they take only the values + 1 and − 1 and are therefore suitable for the binary digital computer.
Abstract: THIS article suggests that Walsh functions1–3 might be used in transform Spectroscopy4–6 to replace the sinusoidal functions appearing in the Fourier transform. We think this might be the case because Walsh functions are a complete orthonormal set, and therefore give rise to an integral transform of Fourier type; and they take only the values + 1 and − 1 and are therefore likely to be well suited to the binary digital computer.
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: 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: A fast method of generating bit-reversed addresses for the fast Fourier transform is described and it is shown that this method can be very fast and efficient.
Abstract: A fast method of generating bit-reversed addresses for the fast Fourier transform is described.
01 Jan 1969
TL;DR: In this paper, an analysis of the fixed-point accuracy of the powqer of two, fast Fourier transform algorithm is presented, leading 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 powqer 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.
TL;DR: A technique for application of the popular fast Fourier transform (FFT) to the system identification problem and an iterative technique is discussed to avoid problems due to the circular nature of convolutions computed by the discrete Fouriertransform (DFT).
Abstract: A technique for application of the popular fast Fourier transform (FFT) to the system identification problem is outlined. Smoothing is obtained inherently in the transform and additionally by redundancy in the data. An iterative technique is discussed for the case of nonzero initial conditions and to avoid problems due to the circular nature of convolutions computed by the discrete Fourier transform (DFT).
TL;DR: In this article, a fast Fourier transform technique was applied to the problem, reducing the required time by more than an order of magnitude. But this technique is not suitable for transducer resolution in boundary layer turbulence.
Abstract: The calculation of the response of a distributed system to a homogeneous random field involves the calculation of an integral of the form ∫S(x − x′, ω)G(x0, x,ω)G*(x0,x′,ω)dxdx′, where S(x − x′, ω) is the cross spectrum of the random field, and G(x0, x,ω) is the system response function. Transformation to the spectral domain, (k − ω space), simplifies the integration and allows considerable savings in computational time. The fast Fourier transform technique can be readily applied to the problem, reducing the required time by more than an order of magnitude. An example of transducer resolution in boundary layer‐turbulence is presented.
TL;DR: In this paper, an efficient method of numerical single and double Fourier transform and inversion with the aid of the complex Fourier series technique coupled with the Cooley-Tukey algorithm for evaluation of the Fourier coefficients is presented.
Abstract: The stochastic structural dynamic analysis for complex structures demands an analytical treatment involving nonstationary stochastic processes, under nonstationary gusty winds or earthquake accelerations. It is emphasized that the frequency domain analysis of such nonstationary problems with the aid of the Fourier transform technique is of vital importance in evaluating the mean value and the covariance function of the response, which are closely related to structural safety. An efficient method of numerical single and double Fourier transform and inversion with the aid of the complex Fourier series technique coupled with the Cooley-Tukey algorithm for evaluation of the Fourier coefficients is presented. Numerical examples indicate that the numerical inversion can be performed within a reasonable time on an IBM 7094.