Showing papers on "Non-uniform discrete 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: 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
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.
77 citations
TL;DR: A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic, and it is found experimentally that if one truncates, rather than rounds, the results of floating point additions and multiplications, the output noise increases significantly.
Abstract: A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic. The result, derived for the case of white input signal, is that the ratio of mean-squared output noise to mean-squared output signal varies essentially as
u = \log_{2}N where N is the number of points transformed. This predicted result is significantly lower than bounds previously derived on mean-squared output noise-to-signal ratio, which are proportional to ν2. The predictions are verified experimentally, with excellent agreement. The model applies to rounded arithmetic, and it is found experimentally that if one truncates, rather than rounds, the results of floating point additions and multiplications, the output noise increases significantly (for a given ν). Also, for truncation, a greater than linear increase with ν of the output noise-to-signal ratio is observed; the empirical results seem to be proportional to ν2, rather than to ν.
75 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.
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.
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.
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, 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: 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.
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.
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
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.
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 paper, the problem of estimating physical spectra from experimental pulse-height distributions is analyzed by means of Fourier transform techniques, and the existence of optimum numbers of degrees of freedom to be assigned to the input data and the spectral estimate is established.
01 Jan 1969
01 Apr 1969
TL;DR: A technique is presented which utilizes the FFT and its associated computational speed, and computes the Fourier transform of "smooth" functions with better accuracy than the F FT alone.
Abstract: The fast Fourier transform (FFT) is a high-speed technique for computing the discrete Fourier transform of a function. The FFT is exact only for discrete (sampled) functions. A technique is presented which utilizes the FFT and its associated computational speed, and computes the Fourier transform of "smooth" functions with better accuracy than the FFT alone. In particular, algorithms using the FFT for transformation of piecewise polynomial functions are presented.
TL;DR: This paper addresses the problem of forming sequential gain and phase estimates needed to permit direct study of time variations in human response and introduces the Gabor (Gaussianweighted Fourier) transform.
Abstract: This paper addresses the problem of forming sequential gain and phase estimates needed to permit direct study of time variations in human response. The conventional Fourier transform with "boxcar" data window is shown to be unsatisfactory. Gabor's theory of elementary signals is cited to show that Fourier transformation with Gaussian data weighting yields an optimum combination of spectral and time resolution. For this window the estimation procedure is constrained by the fundamental relationship ?? . ?t = ? where ??, ?t are the standard deviations of weights across the spectral and data windows, respectively. The Gabor (Gaussianweighted Fourier) transform is introduced. Some consequences of implementing this procedure are briefly discussed and empirical results are presented in verification.
01 Jan 1969