Showing papers on "Discrete-time 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
IBM1
TL;DR: The problem of establishing the correspondence between the discrete transforms and the continuous functions with which one is usually dealing is described and formulas and empirical results displaying the effect of optimal parameters on computational efficiency and accuracy are given.
251 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
IBM1
TL;DR: A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described and some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.
Abstract: A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described. In this method, the normalized phase function for scattering is represented as a Fourier series whose maximum required number of terms depends upon the zenith angles of the directions of incident and of scattered radiation. Some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.
159 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
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: In this article, it is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated.
Abstract: The topics of orthogonality and Fourier series occupy a central position in analysis. Nevertheless, there is surprisingly little statistical literature, with the exception of that of time series and regression, which involves Fourier analysis. In the last decade however, several papers have appeared which deal with the estimation of orthogonal expansions of distribution densities and cumulatives. Cencov [1] and Van Ryzin [6] considered general properties of orthogonal expansion based density estimators and the latter applied these properties to obtain classification procedures. Schwartz [3] and the authors [2] and [4] investigated respectively the Hermite and Trigonometric special cases. The authors also obtained certain general results which apply not only to estimators of the population density but also to estimators of the population cumulative [2], [4] and [5]. In this paper several results derived for the univariate case are extended to the multivariate case. Also a new relationship is obtained which involves general Fourier expansions and estimators. Although there is some reason for calling the Gram-Charlier estimation of distribution densities a Fourier method, one fundamental aspect of Fourier methods is not shared by Gram-Charlier estimation. Gram-Charlier techniques make no use of Parseval's Formula or related error relationships of Fourier analysis. The ease with which the mean integrated square error (MISE) is evaluated, when Fourier methods are applied, accounts for most of the recent interest in this area. Section 1 of this paper deals with an investigation of two general MISE relationships for multivariate estimates of Fourier expansions. The relationship given in Theorem 2 is particularly simple and yet includes the four MISE's which are involved in the estimation problem. In Section 2 the choice of orthogonal functions is restricted to the trigonometric polynomials. It is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated. This result is of considerable utility since it yields a rule for deciding which terms should be included in the estimate of the multivariate density.
73 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
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: In this paper, the M6bius inversion technique is applied to the Poisson summation formula, which results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series.
Abstract: The M6bius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis for the numerical evaluation of sets of Fourier coefficients. The organization of such a calculation is described, and discussed in the context of a broad comparison between this approach and various other standard methods.
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.
TL;DR: A recent paper by Crimmins et al. deals with minimization of mean-square error for group codes by the use of Fourier transforms on groups and a method for representing the groups in a form suitable for machine calculation is shown.
Abstract: A recent paper by Crimmins et al. deals with minimization of mean-square error for group codes by the use of Fourier transforms on groups. In this correspondence a method for representing the groups in a form suitable for machine calculation is shown. An efficient method for calculating the Fourier transform of a group is also proposed and its relationship to the fast Fourier transform is shown. For groups of characteristic two, the calculation requires only N \log_2 N additive operations where N is the order of the group.
TL;DR: Dduce an average purine-pyrimidine pair for DNA using the International Tables for Crystallography to reach the same conclusions as when B = 6 A2.
Abstract: duce an average purine-pyrimidine pair for DNA. The atomic form factors are from International Tables for Crystallography; they are multiplied by weighting factors of 1⁄2/2 or 1⁄44 where appropriate. The temperature factor has B =6A2 since this is more appropriate for DNA. (Those familiar with the diffraction from DNA models are aware that much of the sparseness of DNA data with periodicities less than 3 A is not due only to attenuation factors that increase continuously with diffraction angle but to the fact that the molecular transform itself is small between 3 and 2 A.) However we have repeated the experiments with B = 15 A and 30 A and reach the same conclusions as when B = 6 A2. 9. M. H. F. Wilkins, in Biological Structure and Function (Academic Press, New York, 1960), vol. 1, pp. 13-32. 10. D. A. Marvin, M. H. F. Wilkins, L. D. Hamilton, Acta Cryst. 20, 663 (1966).
TL;DR: The analysis of arbitrary time samples of signals of interest in terms of a Fourier series in effect forces the signal to be periodic with a fundamental period equal to the sample length.
Abstract: The analysis of arbitrary time samples of signals of interest in terms of a Fourier series in effect forces the signal to be periodic with a fundamental period equal to the sample length. This causes sinusoidal components in the signal that are not harmonic in the sample interval to appear to be discontinuous at the ends of the periods; each such component leads to a complete set of the harmonic terms determined by the analysis. The determination of the inharmonic sinusoidal components can be improved by taking suitable combinations of the coefficients determined by the analysis, or by a weighting of the input data to remove the discontinuity. It is shown that improvements of the convergence are accompanied by a corresponding broadening of the principal response.
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: New and simple derivations for the two basic FFT algorithms are presented that provide an intuitive basis for the manipulations involved and reduce the operation to the calculation of a large number of simple two-data-point transforms.
Abstract: The fast Fourier transform (FFT) provides an effective tool for the calculation of Fourier transforms involving a large number of data points. The paper presents new and simple derivations for the two basic FFT algorithms that provide an intuitive basis for the manipulations involved. The derivation for the "decimation in time" algorithm begins with a crude analysis for the zero frequency and fundamental components using only two data samples, one at the beginning and the second at the midpoint of the period of interest. Successive interpolations of data points midway between those previously used result in a refinement of the amplitudes already determined and a first value for the next higher order coefficients. The derivation of the "decimation in frequency" algorithm begins by resolving the original data set into two new data sets, one whose transform includes only even harmonic terms and a second whose transform includes only odd harmonic terms. Since the first of the two new data sets repeats after the midpoint, it can be transformed using only the first half of the data points. The second of the new data sets is multiplied by the negative fundamental function, thereby reducing its order by one and converting it into a data set that transforms into even harmonics only; in this form it can also be transformed using only the first half of the data set. Successive applications of this procedure result finally in reducing the operation to the calculation of a large number of simple two-data-point transforms.
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.
01 Nov 1970
TL;DR: A method is proposed for computing the discrete Fourier transform of complex data whose real and imaginary parts are represented as voltages and operational amplifiers and resistors are the only computing elements required.
Abstract: A method is proposed for computing the discrete Fourier transform of complex data whose real and imaginary parts are represented as voltages. Operational amplifiers and resistors are the only computing elements required.
TL;DR: In this paper, the Fourier transform of the reciprocal of the RPA static longitudinal dielectric constant is approximated by a successive approximation technique, and simple forms are presented for the structure detail of charge screening in an electron gas.
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.