Showing papers on "Fast Fourier transform published in 1969"
TL;DR: This article is intended as a primer on the fast Fourier transform, which has revolutionized the digital processing of waveforms and is needed for a whole new range of applications for this classic mathematical device.
Abstract: For some time the Fourier transform has served as a bridge between the time domain and the frequency domain. It is now possible to go back and forth between waveform and spectrum with enough speed and economy to create a whole new range of applications for this classic mathematical device. This article is intended as a primer on the fast Fourier transform, which has revolutionized the digital processing of waveforms. The reader's attention is especially directed to the IEEE Transactions on Audio and Electroacoustics for June 1969, a special issue devoted to the fast Fourier transform.
668 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: 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: Applications discussed include: enhancement of poles in spectral analysis, high resolution narrow-band frequency analysis, interpolation of band-limited waveforms, and the conversion of a base 2 fast Fourier transform program into an arbitrary radix fast Fouriers transform program.
Abstract: We discuss a computational algorithm for numerically evaluating the z-transform of a sequence of N samples. This algorithm has been named the chirp z-transform algorithm. Using this 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; 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. Applications discussed include: enhancement of poles in spectral analysis, high resolution narrow-band frequency analysis, interpolation of band-limited waveforms, and the conversion of a base 2 fast Fourier transform program into an arbitrary radix fast Fourier transform program.
349 citations
01 Jan 1969
TL;DR: In this article, an analysis of the fixed-point accuracy of the sequence 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 sequence 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 simj=O (1) ulated fixed-point machine and their comparison with the error upper n = 0, 1, * ' * , N - 1.
155 citations
01 Apr 1969
TL;DR: A fast folding algorithm is described which greatly facilitates the correlation of digital data with impulse trains and is useful for detecting weak, noisy pulse trains of unknown period and phase.
Abstract: A fast folding algorithm is described which greatly facilitates the correlation of digital data with impulse trains. This algorithm is useful for detecting weak, noisy pulse trains of unknown period and phase.
96 citations
TL;DR: A transformation that can be used to characterize patterns independent of their position is described, which is 10–100 times faster than the fast Fourier transform (depending on the number of sampling points).
Abstract: The paper describes a transformation that can be used to characterize patterns independent of their position. Examples of the application of the transform for the machine recognition of letters are discussed. The program succeeded in a recognition rate of 80–100% for letters having different position, distortions, inclination, rotation up to 15° and size variation up to 1:3 relative to a reference set of 10 letters. Results with a program for the autonomous learning of new varieties of a pattern (using a learning matrix as an adaptive classifier) are given. When executed on a digital computer, this transform is 10–100 times faster than the fast Fourier transform (depending on the number of sampling points).
92 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
TL;DR: Cape Kennedy vertical wind speed profiles measured by spectral analysis using fast Fourier transform method using spectral analysis method as mentioned in this paper, was used to measure the vertical wind speeds at the Cape Kennedy Airport.
Abstract: Cape Kennedy vertical wind speed profiles measured by spectral analysis using fast Fourier transform method
71 citations
TL;DR: The problems associated with implementing the FFT algorithm in hardware and many of the design options applicable to an FFT processor are described, and a brief comparison of several machine organizations is given.
Abstract: This discussion served as an introduction to the Hardware Implementations Session of the IEEE Workshop on Fast Fourier Transform Processing. It introduces the problems associated with implementing the FFT algorithm in hardware and provides a frame of reference for characterizing specific implementations. Many of the design options applicable to an FFT processor are described, and a brief comparison of several machine organizations is given.
64 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.
TL;DR: A list of features that serves to characterize an FFT processor was developed during a working session of the IEEE Workshop on Fast Fourier Transform Processing and a copy of the table shown in this paper was sent to companies with the request that each company specify the data relating to their machine in the form they would like it to be printed.
Abstract: A list of features that serves to characterize an FFT processor was developed during a working session of the IEEE Workshop on Fast Fourier Transform Processing. Based on this list, information was requested on all of the processors that were known to the Workshop Organizing Committee. A copy of the table shown in this paper was sent to 20 companies with the request that each company specify the data relating to their machine in the form they would like it to be printed. The table represents a compilation of the returns.
21 Nov 1969
TL;DR: In this article, the effects of quantization on implementations of two basic algorithms of digital filtering, the first-or second-order linear recursive difference equation, and the fast Fourier transform (FFT), are studied in some detail.
Abstract: : Quantization effects in digital filters can be divided into four main categories: quantization of system coefficients, errors due to A-D conversion, errors due to roundoffs in the arithmetic, and a constraint on signal level due to the requirement that overflow must be prevented in the comparison. The effects of quantization on implementations of two basic algorithms of digital filtering-the first-or second-order linear recursive difference equation, and the fast Fourier transform (FFT) - are studied in some detail. For these algorithms, the differing quantization effects of fixed point, floating point, and block floating point arithmetic are examined and compared. The ideas developed in the study of simple recursive filters and the FFT are applied to analyze the effects of coefficient quantization, roundoff noise, and the overflow constraint in two more complicated types of digital filters - frequency sampling and FFT filters. Realizations of the same filter design, by means of the frequency sampling and FFT methods, are compared on the basis of differing quantization effects. All the noise analyses in the report are based on simple statistical models for roundoff and A-D conversion errors. Experimental noise measurements testing the predictions of these models are reported, and the empirical results are generally in good agreement with the statistical predictions.
TL;DR: This correspondence presents a mechanization of the fast Fourier transform which results in a particularly simple and compact FORTRAN program without the need for sorting the answers.
Abstract: This correspondence presents a mechanization of the fast Fourier transform which results in a particularly simple and compact FORTRAN program without the need for sorting the answers.
TL;DR: The use of the halftone plotter has permitted us to simplify the theoretical background of the synthesis of a computer-generated hologram, and any spatial filters can be made by computer processing.
Abstract: The study of digital optical information processing by a computer has been performed. As the experimental device, a halftone plotter has been developed which is able to plot the data obtained from an electronic computer with continuous-tone on a conventional cathode ray tube. The capability of producing a halftone drawing by this plotter is useful for displaying optical information processed by a computer. The digital optical information processing studied here is based on this continuous-tone drawing and the reduction of the amount of computer time for the Fourier transform by the new algorithm of the fast Fourier transform. The fast Fourier transform has made it possible to compute in a short time the Fourier spectrum of the object, the convolution integral, the correlation function, etc., by even a middle class computer. These techniques have been applied to some examples. The simulation of filtering and the image restoration by a computer are attempted. The digitized holograms are then synthesized by a computer and are displayed by this plotter on a cathode ray tube as digitized halftone patterns. These holograms are recorded on films and are photoreduced. An image reconstruction is achieved optically for both black-and-white objects and the halftone object. The use of the halftone plotter has permitted us to simplify the theoretical background of the synthesis of a computer-generated hologram. Furthermore, any spatial filters can be made by computer processing.
01 Jan 1969
TL;DR: The problems associated with implementing the FFT algorithm in hardware and many of the design options applicable to an FFT processor are described, and a brief comparison of several machine organizations is given.
Abstract: This discussion served as an introduction to the Hardware Implementations session of the IEEE Workshop on Fast Fourier Transform Processing. It introduces the problems associated with implementing the FFT algorithm in hardware and provides a frame of reference for characterizing specific implementations. Many of the design options applicable to an FFT processor are described, and a brief comparison of several machine organizations is given.
01 Jan 1969
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 essentiallay as ν = log2N, 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.
TL;DR: A fast Fourier transform (FFT) algorithm is presented for an unstructured, parallel ensemble of computing elements with global control that makes efficient use of a fixed-size memory and minimizes data transmission between computing elements.
Abstract: A fast Fourier transform (FFT) algorithm is presented for an unstructured, parallel ensemble of computing elements with global control. The procedure makes efficient use of a fixed-size memory and minimizes data transmission between computing elements. Included are some practical considerations of the trade-offs between element utilization and gain of computing speed via parallelism.
TL;DR: A discrete analog of the narrow-band heterodyne correlator is presented and modified such as to lead to an efficient technique for wide-band FM where several time-compressed references are required.
Abstract: The fast Fourier transform is employed in the design of replica correlation algorithms for application in narrow-band and wide-band active FM sonar. A discrete analog of the narrow-band heterodyne correlator is presented and modified such as to lead to an efficient technique for wide-band FM where several time-compressed references are required.
TL;DR: This paper presents an alternate approach to this problem based on the direct product of matrices, easily understood by anyone familiar with matrix theory, and it yields results in a form convenient for implementation and generalization.
Abstract: Posner (1968) has recently discussed a decoding scheme for certain orthogonal and biorthogonal codes which is based on the fast Fourier transform on a finite abelian group. In this paper, we present an alternate approach to this problem based on the direct product of matrices. This approach is easily understood by anyone familiar with matrix theory, and it yields results in a form convenient for implementation and generalization.
TL;DR: The FFT computing algorithm, when implemented on a modern digital process computer, provides nearly realtime power spectrum analysis of noise signals, making its use attractive for increasing the safety and availability of complex and expensive reactor systems through on-line monitoring of fluctuations in neutron or gamma flux, system pressure, or mechanical vibration.
Abstract: The FFT computing algorithm, when implemented on a modern digital process computer, provides nearly realtime power spectrum analysis of noise signals. Programmable selection of frequency range, filter center frequencies, and bandwidths makes its use attractive for increasing the safety and availability of complex and expensive reactor systems through on-line monitoring of fluctuations in neutron or gamma flux, system pressure, or mechanical vibration.
TL;DR: In this paper, the nonlinear differential equations of circuit analysis are formulated as Volterra integral equations, and the fast Fourier transform may be used to obtain a solution, particularly when the steady-state response to a periodic driving force is required.
Abstract: If the nonlinear differential equations of circuit analysis are formulated as Volterra integral equations, the fast Fourier transform may be used to obtain a solution. The method is faster and more powerful than existing techniques, particularly when the steady-state response to a periodic driving force is required.
TL;DR: In this paper, the equivalence of wave vector filtering and a special form of convolution filtering is demonstrated, and a computationally convenient form of the two-dimensional fast Fourier transform is given for arrays with side lengths not restricted to powers of two.
Abstract: Spatial data can be represented in the two‐ dimensional frequency (wave‐vector) domain. This much simplifies the achievement of any desired transfer response in a linear digital filter. Similar results have been obtained approximately by convolution (also termed correlation) in space with a weighted coefficient set. The equivalence of wave‐vector filtering and a special form of convolution filtering is demonstrated. Given the availability of a fast Fourier transform, computational advantage lies with the former. The regional trend is removed from data prior to wave‐vector filtering, and this combined procedure gives one solution to problems arising at the edge of the data. In the appendix, a computationally convenient form of the two‐dimensional fast Fourier transform is given for arrays with side lengths not restricted to powers of two. Considerable savings in computing time and storage allocation are shown when real data are used.
IBM1
TL;DR: The history of the fast Fourier transform of Cooley and Tukey, as well as experience with general-purpose optimization programs, suggests that publication alone does not result in wide use of a new and vastly more efficient computational technique.
Abstract: The history of the fast Fourier transform of Cooley and Tukey, as well as experience with general-purpose optimization programs, suggests that publication alone does not result in wide use of a new and vastly more efficient computational technique. Recounting his role as entrepreneur and missionary in connection with the fast Fourier transform, the author emphasizes these difficulties and notes the need for mechanisms for easing the cross-utilization of valuable new techniques.
TL;DR: In this article, a quadrature sampling technique is developed that permits efficient coefficient representation of narrow-band signals, and a spatially distributed target composed of Gaussian scatterers is used to derive optimum detection and classification techniques.
Abstract: Utilization of the fast Fourier transform to detect and classify signals generated in the underwater acoustic channel is considered. A quadrature sampling technique is developed that permits efficient coefficient representation of narrow-band signals. A spatially distributed target composed of Gaussian scatterers is used to derive optimum detection and classification techniques. The optimum detector for spatially distributed targets is presented. In addition, cepstrum analysis is employed to extract significant classification clues such as target size and highlight structure. Typical experimental results of target echo, envelope, spectrum, and cepstrum are presented.
IBM1
TL;DR: An associative memory is proposed as a parallel processing unit for the fast Fourier transform; such a processing unit is well suited to implementation in large-scale integrated circuit technologies.
Abstract: An associative memory is proposed as a parallel processing unit for the fast Fourier transform; such a processing unit is well suited to implementation in large-scale integrated circuit technologies. Formulas are derived for the number of memory operations required to execute the algorithm and are tabulated for a range of the number of data points being transformed. It is shown that a 1024 word by 64 bit memory with an operation time of 100 ns could execute a 1024 point transform in 8.4 ms.
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: 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.
01 Jan 1969
TL;DR: In this article, a quadrature sampling technique is developed that permits efficient coefficient representation of narrow-band signals, and a spatially distributed target composed of Gaussian scatterers is used to derive optimum detection and classification techniques.
Abstract: Utilization of the fast Fourier transform to detect and classify signals generated in the underwater acoustic channel is considered. A quadrature sampling technique is developed that permits efficient coefficient representation of narrow-band signals. A spatially distributed target composed of Gaussian scatterers is used to derive optimum detection and classification techniques. The optimum detector for spatially distributed targets is presented. In addition, cepstrum analysis is employed to extract signikant classification clues such as target size and highlight structure. Typical experimental results of target echo, envelope, spectrum, and cepstrum are presented.
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.