Showing papers on "Non-uniform discrete 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
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: 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
44 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
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.
27 citations
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.
27 citations
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
19 citations
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.
13 citations
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.
Patent•
29 Dec 1971
TL;DR: In this article, a digital computer device for computing an approximation of the Fourier transform of a sequence of m sample pairs having known times of occurrence is presented, and an equation expressing the transform is programmed into the computer device.
Abstract: A digital computer device for computing an approximation of the Fourier transform of a sequence of m sample pairs having known times of occurrence. An equation expressing the transform is programmed into the computer device. Binary words representing multiplier coefficients are placed in a first cyclically read memory. The m sample pairs are coded into p-digit words in a second memory from which they are simultaneously available. At each first memory reading 2 m multiplier coefficients are applied at the same time as the 2 m sample binary words to 4 m multipliers. The p-digit multiplier outputs are applied to a plurality of binary adders to perform the algebraic additions required by the equation. The device is particularly adapted for computation of Fourier transform approximations in a received signal processing system for a coherent pulse Doppler radar.
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: 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.
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.
TL;DR: In this paper, a time-varying Fourier transform is defined along with its related power and phase spectra and a convenient recursive technique to compute this transform is also presented.
Abstract: A time-varying Fourier transform is defined along with its related power and phase spectra. A convenient recursive technique to compute this transform is also presented.
TL;DR: In this paper, the convergence criterion of the conjugate series of a Fourier series was deduced by using the second theorem of the Tauber Second Theorem, which is used in this paper.
Abstract: In this paper we establish (c, 1) summability of the sequence {nBn (x)} and by using Tauber's Second Theorem, we deduce the convergence criterion of the conjugate series of a Fourier series.
TL;DR: In this article, the authors define a discrete analog of the classical Fourier transform and present some properties of its properties, including a convolution theorem and an analog of Parseval's indentity.
TL;DR: Simple lenses can be used to produce the Fourier transform of a spatial signal and the use of the experiments in undergraduate engineering laboratories at Oakland University is described.
Abstract: Simple lenses can be used to produce the Fourier transform of a spatial signal. Two optical experiments related to Fourier analysis are described. The first experiment measures the power spectrum of a spatial signal and displays the result directly on an oscilloscope. The second experiment measures both the convolution and the correlation of two spatial signals and again displays the result on an oscilloscope. The use of the experiments in undergraduate engineering laboratories at Oakland University is described.
01 Feb 1971
TL;DR: In this paper, the Fourier transform T:Ll(G)->Co(G) is off if and only if G is finite, which is the same result of Rajagopalan and Segal.
Abstract: A well-known result of Henry Helson is used to prove that a locally compact abelian group is finite if and only if the Fourier transform is a surjective map. Let G be a locally compact abelian group (LCAG) with dual group G (see [3] for definitions and basic facts). It has been proved by Segal in [4] and Rajagopalan in [2] that the Fourier transform T:Ll(G)—>Co(G) is onto if and only if G is finite. Segal's proof appeals to the principal structure theorem for a LCAG and involves proving the result for groups with special properties and then for their Cartesian products, whereas Rajagopalan uses a theorem of Kakutani and Birkhoff to show that if G is extremally disconnected, then G must be discrete. In this note, we provide a short proof using a well-known result about Helson sets. Definition. A compact subset H of G (G not discrete) is a Helson set if every continuous function on H is the restriction of a Fourier transform. Lemma. Let G be nondiscrete with Haar measure m. If H is a Helson set in G, then m(H) = 0. Proof. Let/ be the characteristic function of H and a —f dm. Then if m(H)5¿0, it follows that a is a nonzero bounded Borel measure on H, and Co(G) is onto if and only if G is finite. Received by the editors March 30, 1970. A MS 1969 subject classifications. Primary 4250, 4252.
TL;DR: In this article, the steady state component of the response of a linear time-invariant system to a periodic signal sampled at a rate which is a multiple of the signal frequency is derived.
Abstract: A Z transform theorem is derived which yields the steady-state component of the response of a linear time-invariant system to a periodic signal sampled at a rate which is a multiple of the signal frequency. Wider applications of the theorem are discussed.
01 Jul 1971