Showing papers on "Non-uniform discrete Fourier transform published in 1983"
TL;DR: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable and promises to speed up Fourier-transform calculations.
Abstract: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.
465 citations
TL;DR: In this paper, a new scheme is presented for the determination of the parameters that characterize a multifrequency signal, where the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation.
Abstract: A new scheme is presented for the determination of the parameters that characterize a multifrequency signal. The essential innovation is that the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation. It is shown that by using the Hanning window for tapering substantial improvements are achieved in the following respects: i) more accurate results are obtained for interpolated frequencies, etc., ii) harmonic interference is much less troublesome even if many tones with comparable strengths are present in the spectrum, iii) nonperiodic signals can be handled without an a priori knowledge of the tone frequencies. The stability of the new method with respect to noise and arithmetic roundoff errors is carefully examined.
440 citations
Book•
11 Feb 1983
TL;DR: This chapter discusses Fourier Series and Fourier Transform Algorithms, Discrete Fourier Transforms, DFT Filter Shapes and Shaping, and Spectral Analysis Using the FFT.
Abstract: Preface. Acknowledgments. List of Acronyms. Notation. Introduction. Fourier Series and Fourier Transform. Discrete Fourier Transforms. Fast Fourier Transform Algorithms. FFT Algorithms That Reduce Multiplications. DFT Filter Shapes and Shaping. Spectral Analysis Using the FFT. Walsh-Hadamard Transforms. The Generalized Transform. Discrete Orthogonal Transforms. Number Theoretic Transforms. Appendix. References. Index.
320 citations
TL;DR: Under certain conditions it is shown that discrete-time sequences carry redundant information which then allow for the detection and correction of errors.
Abstract: The relationship between the discrete Fourier transform and error-control codes is examined. Under certain conditions we show that discrete-time sequences carry redundant information which then allow for the detection and correction of errors. An application of this technique to impulse noise cancellation for pulse amplitude modulation transmission is described.
185 citations
Book•
01 Jan 1983
64 citations
TL;DR: It is shown that the self-sorting variants of the mixed-radix FFT algorithm may be specialized to the case of real or conjugate-symmetric input data, and a multiple real/half-complex transform package on the Cray-1 achieves a 30% saving in CPU time compared with a package using conventional algorithms.
53 citations
TL;DR: The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer.
Abstract: The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer. For emission line data, the MEM process is directly used with the interferogram data in place of the regular Fourier transformation process required in Fourier transform spectroscopy. It produces a spectral estimate with an enhanced resolution. For absorption data with a broad background spectrum, the method is applied to a modified interferogram which corresponds to the Fourier transform of the absorptance spectrum. Two results are presented to demonstrate the power of the technique: for the visible emission spectrum of a spectral, calibration lamp and for the infrared chloroform absorption spectrum. Included in the paper is a discussion of the problems associated with practical use of the MEM.
53 citations
TL;DR: In this article, it was shown that the set of signals whose z transform is reducible is contained in the zero set of a certain multidimensional polynomial, and the zero-measure property is obtained as a simple byproduct.
Abstract: The problem of Fourier-transform phase reconstruction from the Fourier-transform magnitude of multidimensional discrete signals is considered. It is well known that, if a discrete finite-extent n-dimensional signal (n ≥ 2) has an irreducible z transform, then the signal is uniquely determined from the magnitude of its Fourier transform. It is also known that this irreducibility condition holds for all multidimensional signals except for a set of signals that has measure zero. We show that this uniqueness condition is stable in the sense that it is not sensitive to noise. Specifically, it is proved that the set of signals whose z transform is reducible is contained in the zero set of a certain multidimensional polynomial. Several important conclusions can be drawn from this characterization, and, in particular, the zero-measure property is obtained as a simple byproduct.
41 citations
Book•
01 Jan 1983
35 citations
TL;DR: New algorithms for DFT(p; k), the discrete Fourier transform on a k-dimensional data set with p points along each array, where p is a prime are exhibited.
Abstract: We exhibit new algorithms for DFT(p; k), the discrete Fourier transform on a k-dimensional data set with p points along each array, where p is a prime. At a cost of additions only, these algorithms compute DFT(p; k) with (pk- 1)/(p - 1) distinct DFT(p; 1) computations.
34 citations
Patent•
07 Jun 1983
TL;DR: In this article, the two-dimensional Fourier transform of the image is interpolated to obtain the values on radial lines, and then the inverse one-dimensional transform is used to reproject the radial lines.
Abstract: Systems and methods are presented for reprojecting images which comprise taking the two-dimensional Fourier transform of the image, interpolating the transform in order to obtain the values on radial lines, and taking the inverse one-dimensional Fourier transforms of the radial lines.
TL;DR: In this article, a new Hankel transform algorithm based on the circular symmetry properties of the input array and two-dimensional vector radix fast-Fourier transform techniques is proposed.
Abstract: The Hankel transform may be defined as the two-dimensional Fourier transform of a circularly symmetric function. A new Hankel-transform algorithm based on this definition is described. The proposed algorithm efficiently generates a rectangularly sampled two-dimensional output array by using the circular symmetry properties of the input array and two-dimensional vector radix fast-Fourier transform techniques. It accomplishes this by partitioning the input matrix into smaller and smaller processing blocks while removing redundant blocks from data manipulations. For applications that require the output data to be sampled on a two-dimensional rectangular raster, the convenience and the computational speed of the resulting algorithm offer advantages over the one-dimensional Hankel-transform algorithms currently available.
TL;DR: In this paper, the effects of additive noise in the given phase on signal reconstruction from the Fourier transform phase are experimentally studied, and the effects on the sequence reconstruction of different methods of sampling the degraded phase of the number of nonzero points in the sequence, and of the noise level, are examined.
Abstract: The effects of additive noise in the given phase on signal reconstruction from the Fourier transform phase are experimentally studied. Specifically, the effects on the sequence reconstruction of different methods of sampling the degraded phase of the number of nonzero points in the sequence, and of the noise level, are examined. A sampling method that significantly reduces the error in the reconstructed sequence is obtained, and the error is found to increase as the number of nonzero points in the sequence increases and as the noise level increases. In addition, an averaging technique is developed which reduces the effects of noise when the continuous phase function is known. Finally, as an illustration of how the results in this paper may be applied in practice, Fourier transform signal coding is considered. Coding only the Fourier transform phase and reconstructing the signal from the coded phase is found to be considerably less efficient (i.e., a higher bit rate is required for the same mean-square error) than reconstructing from both the coded phase and magnitude.
TL;DR: An approximation to the discrete cosine transform (DCT) called the C -matrix transform (CMT) has been developed by Jones et al. as mentioned in this paper for N = 8 and its performance is compared with the DCT based on some standard criteria.
TL;DR: In this article, a new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed, in which all properties of self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform.
Abstract: A new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed. All properties of the self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform. The angular dimensions of the self-images, as well as the notions of the constant of periodical field configuration and the self-image vergence, are introduced. They allow the characterization, in a uniform manner, of the field distribution in the whole space independently of the chosen self-image plane. The equivalency between the self-imaging phenomenon and the image defocusing by an optical system are considered. The general formulae for the harmonics analysis of the intensity distribution are derived.
TL;DR: New recursive techniques for Fourier spectral analysis are reported, for which ongoing spectral estimates are generated from unevenly spaced data in real time, and are particularly attractive in filtering and signal processing applications where signals are not necessarily sampled at a uniform rate.
Abstract: New recursive techniques for Fourier spectral analysis are reported, for which ongoing spectral estimates are generated from unevenly spaced data in real time. The algorithms are robust and computationally efficient, and are well suited to state variable form involving real number calculations. These methods are particularly attractive in filtering and signal processing applications where signals are not necessarily sampled at a uniform rate.
TL;DR: Two VLSI structures for the computation of the discrete Fourier transform are presented; the first is a pipeline working concurrently on different transforms, and it matches within a constant factor the theoretical area-time lower bounds.
Abstract: Two VLSI structures for the computation of the discrete Fourier transform are presented. The first structure is a pipeline working concurrently on different transforms. It is shown that it matches, within a constant factor, the theoretical lower bounds for area versus data rate. The second structure is a simple modification of the first one; it works on a single transform at a time, and it matches within a constant factor the theoretical area-time lower bounds.
01 Oct 1983
TL;DR: It is shown that number theoretic transforms (NTT) can be used to compute discrete Fourier transform (DFT) very efficiently and the total number of real multiplications for a length-P DFT is reduced to (P — 1).
Abstract: Indexing terms: Mathematical techniques, Transforms Abstract: It is shown that number theoretic transforms (NTT) can be used to compute discrete Fourier transform (DFT) very efficiently. By noting some simple properties of number theory and the DFT, the total number of real multiplications for a length-P DFT is reduced to (P — 1). This requires less than one real multiplication per point. For a proper choice of transform length and NTT, the number of shift adds per point is approximately the same as the number of additions required for FFT algorithms.
28 Nov 1983
TL;DR: Discrete fourier transform is represented as a real transform through using number groups and removing redundancy, and is further written in terms of (skew) circular correlations, which can be implemented by fast correlation techniques.
Abstract: Discrete fourier transform is represented as a real transform through using number groups and removing redundancy. The resulting configuration is further written in terms of (skew) circular correlations, which can be implemented by fast correlation techniques. The number of data points considered is a power of 2, even though the method can be generalized to any number of data points.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: This paper investigates the use of polynomial transforms for the implementation of uniform digital bandpass filter banks and shows that this technique reduces significantly the number of arithmetic operations when compared to conventional methods, and yields a regular structure in which most of the computations are performed with FFT-type algorithms.
Abstract: This paper investigates the use of polynomial transforms for the implementation of uniform digital bandpass filter banks. The technique is based upon a decomposition of the N bandpass filters into a set of real polyphase filters followed by a DCT (discrete cosine transform) of size N. The DCT is converted into a DFT (discrete Fourier transform) of size N and the polyphase filters are evaluated by DFT's. This procedure yields a two-dimensional DFT which is computed by a polynomial transform and odd DFT's. We show that this technique reduces significantly the number of arithmetic operations when compared to conventional methods, and yields a regular structure in which most of the computations are performed with FFT-type algorithms.
17 Mar 1983
TL;DR: In this article, the Fourier transform amplitude (magnitude and one bit of phase information) is used to reconstruct a one-dimensional or multi-dimensional sequence from its Fourier Transform amplitude.
Abstract: In this paper, we show that a one-dimensional or multi-dimensional sequence is uniquely specified under mild restrictions by its Fourier transform amplitude (magnitude and one bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multi-dimensional sequence from its Fourier transform amplitude. Reconstruction examples obtained using this algorithm are also provided.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: The mathematical similarities and differences between Fourier transformations and fast Fourier transforms are outlined in this article.
Abstract: Fourier analyses are used in electrophysiological research to reduce EEG data to an interpretable, analyzable form. This article outlines the mathematical similarities and differences between Fourier transforms and fast Fourier transforms. A geometric explanation of the application of fast Fourier transforms and a Fourier series to theta-band EEG data is also included in this article.
TL;DR: In this article, a frequency sampling filter approach is described to compute the discrete Fourier transform (DFT) and the resulting configuration requires delay elements and differential summers which are realizable by simple stray-insensitive switched-capacitor (SC) circuits.
Abstract: Frequency sampling filter approach is described to compute the discrete Fourier transform (DFT). The resulting configuration requires delay elements and differential summers which are shown to be realizable by simple stray-insensitive switched-capacitor (SC) circuits. The proposed scheme finds applications where short data blocks are processed like in radar.
TL;DR: The Fourier transform as discussed by the authors partitions the energy in a waveform into the sum of the energies of simpler components, which is the same as the partitioning of variance into linear contrasts and is a way of measuring the correlation between the waveform and each member of a family of prototype model waveforms.
Abstract: The Fourier transform partitions the energy in a waveform into the sum of the energies of simpler components. This process is the same as the partitioning of variance into linear contrasts and is a way of measuring the correlation between the waveform and each member of a family of prototype model waveforms. Such a partitioning will often, but not always, result in a meaningful decomposition of the original waveform.
TL;DR: In this paper, a computational algorithm for the discrete Fourier transform (DFT) via the discrete Walsh transform (DWT) was proposed, but the calculation equations for the conversion factors from the DWT coefficients to the DFT coefficients have not been shown.
Abstract: We have proposed a computational algorithm for the discrete Fourier transform (DFT) via the discrete Walsh transform (DWT). However, the calculation equations for the conversion factors from the DWT coefficients to the DFT coefficients have not been shown. This paper presents the equations for the conversion factors.
TL;DR: A fast algorithm for an N-point discrete cosine transform (DCT) is derived from a 4N-point Winograd Fourier transform algorithm (WFTA), suitable for a high-speed implementation using one-bit systolic arrays.
Abstract: A fast algorithm for an N-point discrete cosine transform (DCT) is derived from a 4N-point Winograd Fourier transform algorithm (WFTA). This algorithm, which has the same form as Winograd's Fourier transform and convolution algorithms, is suitable for a high-speed implementation using one-bit systolic arrays.
TL;DR: In this article, the apparent Doppler frequency was determined by Fourier transformation of the correlogram followed by two steps of interpolating correction, and the remaining frequency error may be as little as 10-2 j f where integral multiples of j f are the abscissa values for which the discrete Fourier transform is given.
Abstract: Cross-beam laser velocimetry using Bragg shift and photon correlation has been utilized to measure a weak secondary flow component in the presence of strong primary flow at right angles to it. To obtain this velocity component with adequate accuracy, the apparent Doppler frequency was determined by Fourier transformation of the correlogram followed by two steps of interpolating correction. The remaining frequency error may be as little as 10-2 j f where integral multiples of j f are the abscissa values for which the discrete Fourier transform is given. Finally a method is described which allows precise orientation of the cross-beam fringes at right angles to the secondary flow component; it makes use of symmetry properties of the flow in question.
TL;DR: In infrared transmission spectroscopy, several techniques are available for the elimination of interference fringes, including recording polarized spectra of samples mounted at the Brewster angle, selection of experimental conditions so as to preclude the observation of fringes and removal of the signature of the channel spectrum from the Fourier transform of the spectrum as mentioned in this paper.
Abstract: Interference fringes or "channel" spectra are a common problem in infrared transmission spectroscopy. Several techniques are available for the elimination of fringes, including recording polarized spectra of samples mounted at the Brewster angle. subtraction of a sine wave, selection of experimental conditions so as to preclude the observation of fringes, and removal of the signature of the channel spectrum from the Fourier transform of the spectrum.
01 Jan 1983
TL;DR: In this article, a review of the use of newly developed techniques for the analysis of nonlinear data is presented, which are based upon the concept of the spectral transform rather than upon the more traditional Fourier transform.
Abstract: The Fourier transform has historically been the preferred mathematical method for determining analytic solutions to linear wave equations and for analyzing wave data assumed to behave approximately linearly. However, in recent years there has been remarkable progress in the search for exact analytical solutions to certain classes of partial differential equations which describe nonlinear wave evolution. Probably the most successful mathematical approach in this regard has been the spectral or scattering transform which parallels the Fourier method in many important respects. The purpose of this paper is to review the use of newly developed techniques for the analysis of nonlinear data; these methods are based upon the concept of the spectral transform rather than upon the more traditional Fourier transform. We restrict our attention to wave motion governed by the Korteweg-deVries equation on the infinite interval. We give examples of the application of our methods to the analysis of computer generated waveforms, laboratory data and large amplitude internal wave signals obtained in the Andaman Sea.