Showing papers on "Discrete-time Fourier transform published in 1971"
TL;DR: The Fourier transform data communication system is described and the effects of linear channel distortion are investigated and a differential phase modulation scheme is presented that obviates any equalization.
Abstract: The Fourier transform data communication system is a realization of frequency-division multiplexing (FDM) in which discrete Fourier transforms are computed as part of the modulation and demodulation processes. In addition to eliminating the bunks of subcarrier oscillators and coherent demodulators usually required in FDM systems, a completely digital implementation can be built around a special-purpose computer performing the fast Fourier transform. In this paper, the system is described and the effects of linear channel distortion are investigated. Signal design criteria and equalization algorithms are derived and explained. A differential phase modulation scheme is presented that obviates any equalization.
2,507 citations
TL;DR: Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method, and has good potentialities for application in electron microscopy and x-radiography.
Abstract: A new technique is proposed for the mathematical process of reconstruction of a three-dimensional object from its transmission shadowgraphs; it uses convolutions with functions defined in the real space of the object, without using Fourier transforms. The object is rotated about an axis at right angles to the direction of a parallel beam of radiation, and sections of it normal to the axis are reconstructed from data obtained by scanning the corresponding linear strips in the shadowgraphs at different angular settings.
Since the formulae in the convolution method involve only summations over one variable at a time, while a two-dimensional reconstruction with the Fourier transform technique requires double summations, the convolution method is much faster (typically by a factor of 30); the relative increase in speed is larger where greater resolution is required. Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method. It has good potentialities for application in electron microscopy and x-radiography. A new method of reconstructing helical structures by this technique is also suggested.
967 citations
01 Feb 1971
TL;DR: In this paper, a technique is discussed and illustrated for transforming a sequence to a new sequence whose discrete Fourier transform is equal to samples of the z transform of the original sequence at unequally spaced angles around the unit circle.
Abstract: The discrete Fourier transform of a sequence, which can be computed using the fast Fourier transform algorithm, represents samples of the z transform equally spaced around the unit circle. In this letter, a technique is discussed and illustrated for transforming a sequence to a new sequence whose discrete Fourier transform is equal to samples of the z transform of the original sequence at unequally spaced angles around the unit circle.
189 citations
TL;DR: In this paper, the authors consider the special case of a triangle with a vertex at the origin, and assume that the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero.
Abstract: Inequality (1) follows from the special case in which P is a triangle with a vertex at the origin; for any polygon breaks up into triangles, and the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero. Consequently, we can assume P has the form P= {(x, y)C£S\\ (x> y)-t
144 citations
TL;DR: The purpose of this note is to show as clearly as possible the mathematical relationship between the two basic fast methods used for the calculation of discrete Fourier transforms and to generalize one of the methods a little further.
Abstract: The purpose of this note is to show as clearly as possible the mathematical relationship between the two basic fast methods used for the calculation of discrete Fourier transforms and to generalize one of the methods a little further. This method applies to all those linear transformations whose matrices are expressible as direct products.
137 citations
130 citations
TL;DR: In this paper, the authors prove the discrete convolution theorem by means of matrix theory and make use of the diagonalization of a circulant matrix to show that a circular convolution is diagonalized by the discrete Fourier transform.
Abstract: In this paper we prove the discrete convolution theorem by means of matrix theory. The proof makes use of the diagonalization of a circulant matrix to show that a circular convolution is diagonalized by the discrete Fourier transform. The diagonalization of the circular convolution shows that the eigenvalues of a circular convolution operator are identical with the discrete Fourier frequency spectrum.
110 citations
TL;DR: An arbitrary-radix fast Fourier transform algorithm and the design of its implementing signal processing machine are introduced, which yields an implementation with a level of parallelism proportional to the radix r of factorization of the discrete Fouriertransform.
Abstract: An arbitrary-radix fast Fourier transform algorithm and a design of its implementing signal processing machine are introduced. The algorithm yields an implementation with a level of parallelism proportional to the radix r of factorization of the discrete Fourier transform, allows 100 percent utilization of the arithmetic unit, and yields properly ordered Fourier coefficients without the need for pre- or postordering of data.
44 citations
41 citations
TL;DR: In this article, the Fast Fourier Transform (FFT) was used to calculate time-displaced correlation functions from molecular dynamics data much more rapidly (less expansively) than by using the standard integration technique.
40 citations
TL;DR: The design of a class of special-purpose computers for time-series analysis by Fourier transformation is described, which implement machine-oriented fast Fourier transform algorithms obtained by factoring the discrete Fouriertransform to an arbitrary radix.
Abstract: The design of a class of special-purpose computers for time-series analysis by Fourier transformation is described. The computers are sequential machines which implement machine-oriented fast Fourier transform algorithms obtained by factoring the discrete Fourier transform to an arbitrary radix.
01 Oct 1971
TL;DR: An odd discrete Fourier transform (ODFT) which relates in several ways to the usual discrete Fouriers transform (DFT) is introduced and discussed and can readily be applied to spectrum and correlation computations on real signals.
Abstract: An odd discrete Fourier transform (ODFT) which relates in several ways to the usual discrete Fourier transform (DFT) is introduced and discussed. Its main advantage is that it can readily be applied to spectrum and correlation computations on real signals, by halving the storage capacity and greatly reducing the number of necessary steps.
TL;DR: This paper investigates the use of the fast Fourier transform as an aid in the analysis and classification of spectroscopic data and sees how the pattern obtained after transformation is viewed as a weighted average and/or as a frequency representation of the original spectroscopy data.
Abstract: This paper investigates the use of the fast Fourier transform as an aid in the analysis and classification of spectroscopic data. The pattern obtained after transformation is viewed as a weighted average and/or as a frequency representation of the original spectroscopic data. In pattern recognition the Fourier transform allows a different (i.e., a frequency) representation of the data which may prove more amenable to linear separation according to various categories of the patterns. The averaging property means that the information in each dimension of the original pattern is distributed over all dimensions in the pattern resulting from the Fourier transformation. Hence the arbitrary omission or loss of data points in the Fourier spectrum has less effect on the original spectrum. This property is exploited for reducing the dimensionality of the Fourier data so as to minimize data storage requirements and the time required for development of pattern classifiers for categorization of the data. Examples of applications are drawn from low resolution mass spectrometry.
01 Feb 1971
TL;DR: In this paper, a short proof is given of de Leeuw's restriction result for multipliers, and the restriction is shown to hold even for functions with continuous multipliers.
Abstract: A short proof is given of de Leeuw's restriction result for multipliers. In this note we prove directly the following result of de Leeuw (Proposition 3.2 in [l]). Theorem. Let m(x, y) be a Fourier multiplier for Lp(R'+i). Then for almost every x, mx(y)=m(x, y) is a Fourier multiplier for LP(R>) and the multiplier norm of mx does not exceed that of m. In particular, the restriction is possible for each x such that (x, y) is a Lebesgue point of m for almost all y ER'. To prove this we recall that the (necessarily) bounded measurable function m is a Fourier multiplier for Lp(Rn) ii and only if there is a constant C such that, for/, gECÔ(Rn), (*) J m(x)}(x)g(-x)dx = (2^C\\fUg\\p., where/(x) =ff(y)e~ix'vdy and 1/p + l/p' = 1. The best constant C is then the norm of the operator K defined by m}= (Kf)", and we write | m | p for this quantity. Remark. The inequality (*) might be taken as the definition of Fourier multiplier instead of: m(Lv)*Q2(Lv)* for í^pS2, a duality argument for p>2. If p = \ or p = 2 the result is obvious since Fourier transforms of L1 functions restrict and since the Fourier multipliers for L2 are the essentially bounded measurable functions. In the other cases let/, (pECo(Ri), g, ^PECÔ(R'). We assume at first that m is continuous. Apply (*) and Fubini, using f(x)g(y) for f(x, y), tp(x)ip(y) for g(x, y), to deduce that I(x) = ——7 I TM>(x, y)g(y)Ky)dy (lie)' J (2*y is a Fourier multiplier for LP(R'), with |/|pg | wz|p||g||p||v^||P'. Since Received by the editors July 6, 1970. AMS 1969 subject classifications. Primary 4255, 4425.
TL;DR: Continuous and sampled area modulation for single and multichannel operation and hybrid configurations combining area and optical-density modulation are discussed.
Abstract: A one-dimensional function may be represented spatially as an aperture in an opaque screen where the aperture’s width is proportional to the function When such an area-modulated screen is the input to a standard coherent optical fourier transformer, the amplitude of the light along one axis on the output plane is proportional to the fourier transform of the original function Continuous and sampled area modulation for single and multichannel operation are discussed Experimental agreement with the expected results is obtained Hybrid configurations combining area and optical-density modulation are discussed
TL;DR: In this letter, two methods for elim inating the entire zero order are described and the detectability of low spatial frequency components is thereby greatly improved and is now limited only by the background due to scattered light.
Abstract: For coherent optical fourier transformers, aperture tapering (apodization) has been discussed' as a method of improving the detectability of low spatial frequency signals that would other wise be obscured by zero-order light. By zero-order light we mean the light distribution in the transform plane, centered at the zero frequency point, due to the average of dc bias light at the input aperture. In this letter, two methods for elim inating the entire zero order are described. The detectability of low spatial frequency components is thereby greatly improved and is now limited only by the background due to scattered light.
TL;DR: It is shown how the spline transform reduces errors introduced by the discrete transform and alleviates noise problems when the sampling rate is limited due to experimental method or hardware constraints.
Abstract: The transform of a spline-function approximation to continuous data is called a spline transform. In this correspondence, the spline and the discrete Fourier transforms (DFT) are compared as means for numerical computation of the Fourier integral transform. It is shown how the spline transform reduces errors introduced by the discrete transform and alleviates noise problems when the sampling rate is limited due to experimental method or hardware constraints.
TL;DR: In this article, the Fourier components of the density-density correlation function in a fluid obtained from the linearized hydrodynamic equations can also be obtained by adopting a particularly simple form for the associated memory function.
Abstract: It is shown that the expression for the Fourier components of the density-density correlation function in a fluid obtained from the linearized hydrodynamic equations can also be obtained by adopting a particularly simple form for the associated memory function The result is used to calculate the longitudinal viscosity of a fluid in terms of the moments of the space and time Fourier transform of the density-density correlation function S(q, ω)
TL;DR: In this article, the Fourier transform integrals are evaluated using the trapezoidal rule and Filon's Filon-based Fourier Transform Transform (FFT) algorithm, and an algorithm proposed by Cooley and Tukey for doing Fourier sums rapidly.
01 Feb 1971
Abstract: Let G be a compact abelian group. We consider the ideal of functions in LP(G) with unconditionally convergent Fourier series in the Lp norm. This ideal is shown to coincide with the "Derived algebra" of Helgason. A characterization of this ideal is given when p is an even integer. Let G be a compact abelian group with dual group Y and with normalized Haar measure (so that/—»/ is an isometry of L2(G) onto ¿2(r)). For 1 ̂ p ^ », LP(G) is a Banach algebra with convolution for multiplication. Let Sp denote the ideal of functions in Lp having unconditionally convergent Fourier series in the Lv norm. In §2 we show that, for the Banach algebra Lp, Sp coincides with the "Derived algebra" of Helgason [7]; from this and [7] one immediately obtains that SP = L2 for 1 ̂ p<2. This is a special case of a theorem of Grothendieck [4]. In §3 we show that, for p an even integer, if fELp and/èO then then fESp. From this it follows that (for p an even integer) Sp coincides with the set of fELp such that |/| is the transform of an Lp function. Thus for such p one can characterize the Rudin A(p) sets as those subsets ECr for which |/| is the transform of an Lp function whenever/GZp and/= 0 outside of E. 1. Preliminaries. For facts we use about harmonic analysis the reader is referred to [3] and [Q], for unconditional convergence and related notions, to [2], and for LP(G) and dual algebras, to [8]. Let 5 denote the set of finite subsets of Y. Theorem 1. Let i^p^ ». Then Sp is a semisimple dual Banach algebra with the norm given by
TL;DR: In this article, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions / £ C(p>[0, 1], where the length of the interval need not coincide with a period of the trigonometrical weighting function.
Abstract: In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions / £ C(p>[0, 1]. In this part two extensions of the theory are described. One modification extends the theory to piecewise continuous functions, / £ PC(p'[0, 1]. Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the interval need not coincide with a period of the trigonometrical weighting function). The other modification treats functions which are analytic, but whose low-order derivatives vary rapidly due to poles in the complex plane near the interval of integration. Essentially these poles are 'subtracted out' but this is done implicitly by the inclusion of additional terms in the standard series. The practical application of these modified methods requires that the nature and location of the discontinuities—or poles—be known at least approximately.
TL;DR: In this paper, the concept of position spectrum for discrete orthogonal transformations of N-periodic sequences is introduced, and it is shown that a position spectrum is analogous to the conventional Fourier phase spectrum.
Abstract: The concept of "position spectrum" for discrete orthogonal transformations of N-periodic sequences is introduced. It is shown that a position spectrum is analogous to the conventional Fourier phase spectrum. As an illustration, the position spectrum for a modified Hadamard or BIFORE (binary Fourier representation) transform is developed.
TL;DR: It is shown that in computing the spectrum of a bandlimited process, the trapezoidal rule is preferred when judged by the criterion of choosing the integration formula which leads to the coarsest sampling of the data.
Abstract: It is possible to view the discrete Fourier transform as the result of approximating the Fourier integral by a trapezoidal rule integration formula. In this correspondence the effects of using higher ordered Newton–Cotes integration formulas are examined. It is shown that in computing the spectrum of a bandlimited process, the trapezoidal rule is preferred when judged by the criterion of choosing the integration formula which leads to the coarsest sampling of the data.
TL;DR: Some problems associated with Fourier analysis of reaction time data are indicated and convenient ways to overcome some of the difficulties are suggested.
Abstract: The popularity of the assumption of stages in models of the reaction time process and the availability of fast and efficient means of computing approximations to the Fourier transform makes the Fourier analysis of reaction time data attractive. This paper indicates some problems associated with such analyses and suggests convenient ways to overcome some of the difficulties.
TL;DR: In this article, a shipborne wave-recording system consisting of a sonic wave gauge, accelerometers, gyroscopes and a computer system is described, where signals from the measuring apparatus are fed directly into a ship-borne digital computer system at a prescribed sampling rate.
Abstract: A shipborne wave-recording system which consists of a sonic wave gauge, accelerometers, gyroscopes and a computer system is described. Signals from the measuring apparatus are fed directly into a shipborne digital computer system at a prescribed sampling rate. The time series of wave heights and the acceleration are transformed into Fourier series using an algorithm of Fast Fourier Transform. Errors contained in the observed wave heights due to ship motion are corrected in the Fourier series by using the Fourier coefficients for the vertical acceleration. Power spectra and waveforms can also be calculated in a short time with this system from Fourier coefficients. Examples of the observational results obtained in the central part of the East China Sea in 1969 are presented.