Showing papers on "Non-uniform discrete Fourier transform published in 1970"
TL;DR: It is shown that the discrete equivalent of a chirp filter is needed to implement the computation of the discrete Fourier transform (DFT) as a linear filtering process, and that use of the conventional FFT permits the computations in a time proportional to N \log_{2} N for any N.
Abstract: It is shown in this paper that the discrete equivalent of a chirp filter is needed to implement the computation of the discrete Fourier transform (DFT) as a linear filtering process. We show further that the chirp filter should not be realized as a transversal filter in a wide range of cases; use instead of the conventional FFT permits the computation of the DFT in a time proportional to N \log_{2} N for any N, N being the number of points in the array that is transformed. Another proposed implementation of the chirp filter requires N to be a perfect square. The number of operations required for this algorithm is proportional to N^{3/2} .
410 citations
TL;DR: This work reports here on the use of a phase mask which imparts a phase shift of 180 degrees to half the data spots chosen at random and shows that the intensity in the Fourier transform plane is now proportional to the intensity of the Fouriers of one single data spot.
Abstract: In a holographic page-oriented memory the information is stored in an array of holograms. It is advantageous to record the Fourier transform of the original data mask because the minimum space bandwidth is then required and the information about any one data bit is spread over the hologram plane. In the Fourier transform plane most of the light is concentrated in an array of bright “spikes” because the data mask consists of an array of equidistant data spots. Some means is needed to distribute the light more evenly. We report here on the use of a phase mask which imparts a phase shift of 180° to half the data spots chosen at random. An analysis shows that the intensity in the Fourier transform plane is now proportional to the intensity of the Fourier transform of one single data spot.
225 citations
TL;DR: A novel structure for a hardwired fast Fourier transform (FFT) signal processor that promises to permit digital spectrum analysis to achieve throughput rates consistent with extremely wide-band radars is described.
Abstract: This paper describes a novel structure for a hardwired fast Fourier transform (FFT) signal processor that promises to permit digital spectrum analysis to achieve throughput rates consistent with extremely wide-band radars. The technique is based on the use of serial storage for data and intermediate results and multiple arithmetic units each of which carries out a sparse Fourier transform. Details of the system are described for data sample sizes that are binary multiples, but the technique is applicable to any composite number.
127 citations
01 Apr 1970
TL;DR: In this paper, a Hilbert transformation procedure for discrete data has been developed, which is useful in a variety of applications such as the analysis of sampled data systems and the simulation of filters.
Abstract: A Hilbert transformation procedure for discrete data has been developed. This transform could be useful in a variety of applications such as the analysis of sampled data systems and the simulation of filters.
119 citations
TL;DR: This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used, and upper and lower bounds for the total relative meansquare error are given.
Abstract: The fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier coefficients with a substantial time saving over conventional methods. The finite word length used in the computer causes an error in computing the Fourier coefficients. This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used. Upper and lower bounds for the total relative mean square error are given. The theoretical results are in good agreement with the actual error observed by taking the FFT of data sequences.
89 citations
TL;DR: The techniques disclosed here should be especially important in real-time estimation of power spectra, in instances where the data sequence is essentially unterminated.
Abstract: A common application of the method of high speed convolution and correlation is the computation of autocorrelation functions, most commonly used in the estimation of power spectra. In this case the number of lags for which the autocorrelation function must be computed is small compared to the length of the data sequence available. The classic paper by Stockham, revealing the method of high speed convolution and correlation, also discloses a number of improvements in the method for the case where only a small number of lag values are desired, and for the case where a data sequence is extremely long. In this paper, the special case of autocorrelation is further examined. An important simplification is noted, based on the linearity of the discrete Fourier transform, and the circular shifting properties of discrete Fourier transforms. The techniques disclosed here should be especially important in real-time estimation of power spectra, in instances where the data sequence is essentially unterminated.
66 citations
TL;DR: A procedure for factoring of the N×N matrix representing the discrete Fourier transform is presented which does not produce shuffled data, and is shown to be most efficient for Na power of two.
Abstract: A procedure for factoring of the N×N matrix representing the discrete Fourier transform is presented which does not produce shuffled data. Exactly one factor is produced for each factor of N, resulting in a fast Fourier transform valid for any N. The factoring algorithm enables the fast Fourier transform to be implemented in general with four nested loops, and with three loops if N is a power of two. No special logical organization, such as binary indexing, is required to unshuffle data. Included are two sample programs, one which writes the equations of the matrix factors employing the four key loops, and one which implements the algorithm in a fast Fourier transform for N a power of two. The algorithm is shown to be most efficient for Na power of two.
66 citations
IBM1
TL;DR: Alternative methods for the estimation of spectra are described and compared and general questions of statistical variability, the use of regression methods to smooth the periodogram, and use of time sectioning of the data to either smooth or to investigate non-stationarities in the data are discussed.
51 citations
34 citations
TL;DR: Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.
Abstract: This paper is devoted to a discussion of discrete spectrum analysis which is important in applicational areas such as sonar and replica correlation. The discrete Fourier transform is shown to arise naturally as a consequence of finite impulsive sampling and the fast Fourier transform is introduced as the most efficient means of computing the discrete Fourier transform. These are described in terms of parameters pertinent to digital sonar signal processing, including resolution, dynamic range, and processing gain. Computational accuracy is investigated as a function of word lengths associated with the data, kernels, and intermediate transforms for both conditional and automatic array scaling. In real-time equipment, it is frequently necessary to employ some sort of automatic gain control and such a device is investigated here. Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.
Patent•
02 Sep 1970
TL;DR: In this paper, an approach for deriving in essentially real-time unweighted and weighted continuous electrical representations of the Fourier transform and/or the inverse-fourier transform of a complex waveform is presented.
Abstract: Apparatus and methods for deriving in essentially real time unweighted and weighted continuous electrical representations of the Fourier transform and/or the inverse Fourier transform of a complex waveform. In performing the Fourier transform, the input waveform is sampled at the Nyquist sampling rate and the samples stored in respective sample-and-hold circuits. These samples are applied to signal generating circuitry for deriving harmonically related time-varying cosine and sine signals having peak values corresponding to weighted or unweighted values of respective ones of the sample-and-hold circuit outputs, and having a fundamental frequency which may be chosen independently of the frequency content of the input waveform. These cosine and sine signals are then respectively summed for producing resultant summed sine and cosine signals which respectively correspond to weighted or unweighted representations of the real and imaginary components of the Fourier transform of the input waveform with the frequency variable being simulated by time. In one embodiment, these summed sine and cosine signals are applied to a function generator for generating signals representative of the weighted or unweighted amplitude spectrum and/or phase spectrum of the input waveform for further application to appropriately calibrated and adjusted oscilloscopes for producing visual displays thereof. In another embodiment, these resultant summed sine and cosine signals are in turn sampled at the Nyquist sampling rate to provide samples which may conveniently be modified in accordance with desired criteria. The modified samples are then recombined using the inverse Fourier transform technique of the invention which employs circuitry basically similar to that used for the Fourier transform to produce an output signal representative of the original input signal and containing the modifications produced in accordance with the desired criteria.
TL;DR: This paper presents the results of the fast Fourier transform in sufficient detail that interested nonexperts can obtain the computer algorithm, and the necessary label permutations, and points out the well known utility of base 2.
Abstract: The fast Fourier transform is usually described as a factorization. Recently this has been done in matrix terms. In this paper we present these results in sufficient detail that interested nonexperts can obtain the computer algorithm, and the necessary label permutations. We also count the number of arithmetic operations required in the calculation and point out the well known utility of base 2, both because of mathematical and machine hardware considerations. A simple FORTRAN program based on these ideas is included.
TL;DR: It is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.
Abstract: Since controversy has arisen as to whether the Blackman-Tukey or the fast Fourier transform (FFT) technique should be used to compute power spectra, single and cross spectra have been computed by each approach for artificial data and real data to provide an empirical means for determining which technique should be used. The spectra were computed for five time series, two sets of which were actual field data. The results show that in general the two approaches give similar estimates. For a spectrum with a large slope, the FFT approach allowed more window leakage than the Blackman-Tukey approach. On the other hand, the Blackman-Tukey approach demonstrated a better window closing capability. From these empirical results it is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.
01 Jul 1970
TL;DR: In this paper, a fast Fourier transform technique is described for the approximate numerical evaluation of distribution functions directly from characteristic functions, which can be used to estimate the distribution function directly from the characteristic function.
Abstract: A fast Fourier transform technique is described for the approximate numerical evaluation of distribution functions directly from characteristic functions Examples are presented
TL;DR: In this article, the properties of BIFORE (Hadamard) transforms are compared with those of discrete Fourier transform (d.f.t.) and compare their properties to those of the Hadamard transform.
Abstract: Elementary properties of discrete Fourier transforms (d.f.t.) have appeared in recent literature. Corresponding properties of BIFORE (Hadamard) transforms are summarised and compared with those of the d.f.t.
TL;DR: In this article, a complex BIFORE transform is defined and elementary properties of the transform are developed, and the complexity of the transformation is analyzed and the properties of its properties are analyzed.
Abstract: A complex BIFORE transform is defined and elementary properties of the transform are developed.
TL;DR: In this paper, Coooley and Tukey's fast Fourier transform algorithm for two dimensional complex data has been modified so as to reduce the storage space and computation time to half.
Abstract: Cooley andTukey's fast Fourier transform algorithm for two dimensional complex data has been modified so as to reduce the storage space and computation time to half. The modified version has enabled us to Fourier transform aeromagnetic field over twice the area that could be covered by the original method. From the Fourier transform we computed radial spectrum, which could be approximated by three straight line segments whose slopes are related to the depths of the various magnetic layers. The computed depths are: 1090', 2600', and 7200'.
TL;DR: In this paper, Fourier transform holograms of transilluminated objects are considered and it is shown that for a fixed recording configuration, both a given number of resolution cells and a given signal to noise ratio can be obtained in the reconstructed image.
01 Aug 1970
TL;DR: In this article, the N-dimensional discrete Fourier transform (DFT) is represented as a matrix of elements of unit magnitude, with the arguments constructed as inner products of lattice vectors in the sampling and wavenumber domains, filling regions inverse to the basic cells on their respective lattices.
Abstract: The N-dimensional discrete Fourier transform (DFT) may be represented as a matrix of elements of unit magnitude, with the arguments constructed as inner products of lattice vectors in the sampling and wavenumber domains, filling regions inverse to the basic cells on their respective lattices. The fast Fourier transform numerical technique is directly applicable to this configuration.
TL;DR: A discrete Fourier transform method for factoring arbitrary spectral density functions is presented and an expression for the absolute error is presented.
Abstract: A discrete Fourier transform method for factoring arbitrary spectral density functions is presented. The factorization can be implemented in a straightforward and efficient manner, and it does not require that the spectra be rational. An expression for the absolute error is also presented.
TL;DR: In this article, a transform with a square-wave kernel is proposed to complement the fast Fourier transform (f.f.t.) algorithm, being a trapezoidal integration rule, giving errors in the tails of spectra.
Abstract: Unless inconveniently high sampling rates are used, the fast-Fourier-transform (f.f.t.) algorithm, being a trapezoidal integration rule, gives errors in the tails of spectra. A transform with a square-wave kernel, which can be evaluated accurately in a simple manner, is proposed to complement the f.f.t. An example involving measured data is included.
TL;DR: In this article, a convolution with a Gaussian instrumental profile is proposed to eliminate noise and the drawbacks of a discrete Fourier transform, and a numerical filter is introduced, preferably a filter with Gaussian shape.
TL;DR: The Cooley and Tukey algorithm used in the present work is much faster than the usual Fourier method since the length of computation is proportional to N log 2(N) rather than N2 as mentioned in this paper.
Abstract: Recently it has been suggested that fast finite Fourier transforms be employed for the solution of Tung's integral equation. The Cooley and Tukey algorithm used in the present work is much faster than the usual Fourier method since the length of computation is proportional to N log2(N) rather than N2. This saves computer time and also enables a larger number of points to be used in order to facilitate computer plotting of the corrected chromatogram. First some of the basic properties of finite Fourier transform are presented in order to familiarize the reader with the approximations involved. Then several chromatograms, both analytical and simulated experimental are considered and some of the problems inherent in processing experimental chromatograms are discussed.
TL;DR: An algorithm for computing the discrete Walsh transform (abstract Fourier transform) of a sampled periodic function whose domain of definition is the set of integers modulo 2n.
Abstract: J. L. Shanks1has given an algorithm for computing the discrete Walsh transform (abstract Fourier transform) of a sampled periodic function whose domain of definition is the set of integers modulo 2n. An algorithm of the same efficiency, using a much simpler notation, was given for the abstract Fourier transform in my correspondence published in 1963 in this TRANSACrIONS.2This transform has an identical matrix representation; the only difference is that the function domain is represented (for computation purposes) by binary coded representations of the integers from 0 to 2n−1. These binary n-tuples form a group under vector addition, modulo two.
01 Sep 1970
TL;DR: In this article, the extended operator concept is used to introduce artificial periodicity, and a discrete Fourier expansion becomes possible, yielding state-vector equations in the expansion coefficients for distributed systems involving 1st-order spatial operators with 2-point boundary values.
Abstract: Eigenfunction expansion of linear distributed systems is only possible when discrete solutions exist to the spatial-operator eigenvalue problem. 1st-order systems are naturally associated with a continuous spectrum, and a reduction to a finite lumped-system approximation is accordingly difficult. When the extended operator concept is used to introduce artificial periodicity, however, a discrete Fourier expansion becomes possible, yielding state-vector equations in the expansion coefficients. This generalised Fourier method particularly applies to distributed systems involving 1st-order spatial operators with 2-point boundary values.
TL;DR: In this paper the transmission-line equations and their solutions are presented in terms of Fourier transforms, yet provides a tool that can handle any load and source impedance with any Fourier transformable waveform.
Abstract: In this paper the transmission-line equations and their solutions are presented in terms of Fourier transforms. This approach is particularly useful in presenting transmission lines to undergraduates, yet provides a tool that can handle any load and source impedance with any Fourier transformable waveform.
TL;DR: In this paper, a comparison between a system identification method using the fast Fourier transform and one which is optimal is made, and it is shown that for systems with long settling times the transform method possesses computational advantages but at the cost of accuracy.
Abstract: Comparison between a system identification method using the fast Fourier transform and one which is optimal is made. The comparison shows that for systems with long settling times the transform method possesses computational advantages but at the cost of accuracy.