Showing papers on "Non-uniform discrete Fourier transform published in 1988"
TL;DR: In this article, a unitary fast Fourier transform method for solving time dependent curve crossing problems is presented, and the procedure is described in detail and illustrated by calculations for a two curve, one dimensional example.
Abstract: We develop a unitary fast Fourier transform method for solving time dependent curve crossing problems. The procedure is described in detail and is illustrated by calculations for a two curve, one‐dimensional example. The time evolution of the wave function and mean nuclear positions and energies for each curve are shown and discussed.
137 citations
TL;DR: In this paper, two mathematical procedures of band narrowing using Fourier transforms are discussed and the methods of Fourier self-deconvolution and Fourier derivation are easier to use and to control than procedures which rely on the use of convolution functions.
113 citations
TL;DR: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.
Abstract: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary. An important special case also considered is the linear (unattenuated) Radon transform (LRT). The solution is on the form of an orthogonal function expansion matched to projections that are in parallel-ray geometry. This property allows for efficient and accurate processing of the projections with fast Fourier transform (FFT) without interpolation or beam matching. The algorithm is optimized by the use of boundary conditions on the 2-D Fourier transform of the sinogram. These boundary conditions imply that the signal energy of the sinogram is concentrated in well-defined sectors in transform space. The angle defining the sectors depends in a direct way on the radius of the field view. These results are also obtained for fan-beam geometry and the linear Radon transform (the Fourier-Chebyshev transform of the sinogram) to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates. >
107 citations
TL;DR: It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFT’s is equal to thenumber of linear congruences spanning the N-N grid.
Abstract: An algorithm is presented for computation of the two-dimensional discrete Fourier transform (DFT). The algorithm is based on geometric properties of the integers and exhibits symmetry and simplicity of realization. Only one-dimensional transformation of the input data is required. The transformations are independent; hence, parallel processing is feasible. It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFTs is equal to the number of linear congruences spanning the N*N grid. Examples for N=3, N=4, and N=10 are presented. A short APL code illustrating the algorithm is given. >
76 citations
TL;DR: Preliminary results are presented on the VLSI design and implementation of a novel algorithm for accurate high-speed Fourier analysis and synthesis, based on the number-theoretic method of Mobius inversion.
Abstract: Preliminary results are presented on the VLSI design and implementation of a novel algorithm for accurate high-speed Fourier analysis and synthesis. The arithmetic Fourier transform (AFT) is based on the number-theoretic method of Mobius inversion. Its computations proceed in parallel, and the individual operations are very simple. Except for a small number of scalings in one state of the computation, only multiplications by 0, +1, and -1 are required. If the input samples were not quantized and if ideal real-number operations were used internally, then the results would be exact. The accuracy of the computation is limited only by the input A/D (analog-to-digital) conversion process, any constraints on the word lengths of internal accumulating registers, and the implementation of the few scaling operations. Further simplifications are obtained by using delta modulation to represent the input function in digital form, so that only binary (or preferably, ternary) sequences needs to be processed in the parallel computations. The required accumulations can be replaced by up/down counters. The dynamic range of the resulting transformation can be increased by the use of adaptive delta modulation. >
63 citations
TL;DR: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented and is combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two- dimensional DFT.
Abstract: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented. It is then combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two-dimensional DFT. This mixed model takes stream input and produces stream output. In addition, no extra I/O time delay is required before performing the row (column) transform of the two-dimensional DFT. >
55 citations
TL;DR: The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm.
Abstract: A relatively unexplored algorithm is developed for reconstructing a two-dimensional image from a finite set of its sampled projections. The algorithm, referred to as the Hankel-transform-reconstruction (HTR) algorithm, is polar-coordinate based. The algorithm expands the polar-form Fourier transform F(r, theta ) of an image into a Fourier series in theta ; calculates the appropriately ordered Hankel transform of the coefficients of this series, giving the coefficients for the Fourier series of the polar-form image f(p, phi ); resolves this series, giving a polar-form reconstruction; and interpolates this reconstruction to a rectilinear grid. The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm. >
45 citations
Journal Article•
42 citations
TL;DR: In this paper, it was shown that extended zero-filling (e.g., a "zoom" transform) actually reduces the accuracy with which the spectral peak position can be determined, and that the results can be more accurate when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic.
Abstract: A frequency-domain Lorentzian spectrum can be derived from the Fourier transform of a time-domain exponentially damped sinusoid of infinite duration. Remarkably, it has been shown that even when such a noiseless time-domain signal is truncated to zero amplitude after a finite observation period, one can determine the correct frequency of its corresponding magnitude-mode spectral peak maximum by fitting as few as three spectral data points to a magnitude-mode Lorentzian spectrum. In this paper, we show how the accuracy of such a procedure depends upon the ratio of time-domain acquisition period to exponential damping time constant, number of time-domain data points, computer word length, and number of time-domain zero-fillings. In particular, we show that extended zero-filling (e.g., a "zoom" transform) actually reduces the accuracy with which the spectral peak position can be determined. We also examine the effects of frequency-domain random noise and round-off errors in the fast Fourier transformation (FFT) of time-domain data of limited discrete data word length (e.g., 20 bit/word at single and double precision). Our main conclusions are: (1) even in the presence of noise, a three-point fit of a magnitude-mode spectrum to a magnitude-mode Lorentzian line shape can offer an accurate estimate of peak position in Fourier transform spectroscopy; (2) the results can be more accurate (by a factor of up to 10) when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic; and (3) FFT roundoff errors can be made negligible by use of sufficiently large (> 16 K) data sets.
42 citations
TL;DR: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it and their application to discrete number theory is proposed.
Abstract: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it.
39 citations
TL;DR: In this paper, the exact expression for the discrete Fourier transform of a sum of exponentially damped sinusoids is derived, and its applicability for describing the general DFT spectrum is demonstrated.
07 Jun 1988
TL;DR: An efficient method for computing the discrete Fourier transform when only a few output points are needed is described, based on a novel factorization of the DFT, where one part is computed using standard power-of-two FFTs and the other uses a technique similar to the Goertzel algorithm.
Abstract: The authors describe an efficient method for computing the discrete Fourier transform (DFT) when only a few output points are needed. The method is shown to be more efficient than either Goertzel's method or pruning, and it allows any band in the output to be computed. It is based on a novel factorization of the DFT, where one part is computed using standard power-of-two FFTs (fast Fourier transforms) and the other uses a technique similar to the Goertzel algorithm. >
CNET1
TL;DR: The authors indicate an apparently novel method for computing an inverse discrete Fourier transform (IDFT) through the use of a forward DFT program, and point out that, in many cases, this is obtained without any additional cost, either in terms of program length or in Terms of computational time.
Abstract: The authors indicate an apparently novel method for computing an inverse discrete Fourier transform (IDFT) through the use of a forward DFT program. They point out that, in many cases, this is obtained without any additional cost, either in terms of program length or in terms of computational time. >
01 Dec 1988
TL;DR: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform.
Abstract: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform. Compared with existing algorithms, the number of multiplications is reduced by a factor of 8 to 20, at the expense of a slight increase in the number of shift and add operations, that are assumed to be simpler than multiplications.
01 Jan 1988
TL;DR: In this article, the multiplicative complexity of the discrete Fourier transform (DFT) was analyzed and the complexity of DFT for any positive integer was shown. But the complexity was not shown for any integer.
Abstract: In this chapter the multiplicative complexity of the discrete Fourier transform (DFT) is analyzed. The next several sections define the DFT and then show how the complexity of the DFT is determined when the number of inputs is prime, a power of an odd prime, a power of two, and finally for any positive integer.
TL;DR: The coding method used for the comparison utilizes a marginal bit-allocation scheme and Lloyd-Max quantizer as discussed by the authors, and the discrete cosine transform outperforms the other three transforms based on the mean-square quantization error.
Abstract: The coding method used for the comparison utilizes a marginal bit-allocation scheme and Lloyd-Max quantizer. Gamma, Laplacian, and Gaussian models are compared for the distribution of the transform coefficients. The discrete cosine transform outperforms the other three transforms based on the mean-square quantization error. >
TL;DR: It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method.
Abstract: The discrete short-time transform (DSTT) is a generalization of the discrete short-time Fourier transform (DSTFT). The necessary and sufficient conditions on the analysis filter, under which perfect reconstruction of the input signal is possible (when the DSTT is not modified), are presented. The class of linear modifications for which the original input can be reconstructed when the modification is applied is characterized. The synthesis of an optimal (in the minimum-mean-square-error sense) signal from a modified DSTT (MDSTT) of finite duration is presented. It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method. For longer analysis filters, the optimal synthesis scheme becomes quite cumbersome, and therefore, a steady-state solution (as the duration of the MDSTT approaches infinity) is presented for this case. It is shown that this solution can be approximated with arbitrarily small reconstruction error. >
TL;DR: It is shown that the use of equal spacings in the logarithmic time and frequency domains provides a very efficient transform algorithm that is applicable for the analysis of systems with moderate dynamic behavior over several frequency decades.
Abstract: Discrete Fourier transforms are derived which allow the use of nonequally spaced time-domain samples. It is shown that the use of equal spacings in the logarithmic time and frequency domains provides a very efficient transform algorithm. The applicability of this algorithm for the analysis of systems with moderate dynamic behavior over several frequency decades is demonstrated by examples. An error analysis is given. >
TL;DR: Based on a multichannel incoherent optical correlator, a new simple scheme was proposed for performing a complex discrete Fourier transform as mentioned in this paper, where a complex value is represented by using three nonnegative reals, and every real is encoded with the area of a rectangular aperture.
01 Jan 1988
TL;DR: A numerical method for eliminating channel spectra from data obtained with a bandlimited infrared Fourier transform spectrometer is developed and is compared with others in common use on a synthetic model spectrum.
Abstract: A numerical method for eliminating channel spectra from data obtained with a bandlimited infrared Fourier transform spectrometer is developed. The method is compared with others in common use on a synthetic model spectrum.
IBM1
TL;DR: In this article, the authors obtained explicit formulas for the probability distributions of such bursts and for the errors that the clipping induce in the decoder, which can help the FTDM code designer to decide on an appropriate average power constraint.
Abstract: Inherent in the method of Fourier transform division multiplexing (FTDM) is the possibility that the FTDM encoder will yield spurious power bursts, which can affect the linearity of the channel. A common way for dealing with such bursts is to clip the signal at some predetermined peak power level. The authors obtain explicit formulas for the probability distributions of such bursts and for the errors that the clipping induce in the decoder. The formulas can help the FTDM code designer to decide on an appropriate average power constraint. >
20 Mar 1988
TL;DR: In this paper, the authors examined the effect of these dangers when the FFT algorithm is supplied to power system load variation and recommended the use of a DFT algorithm to evaluate the frequency spectrum of power system loads.
Abstract: The discrete Fourier transform (DFT) and the fast Fourier transform (FFT) are based on certain assumptions that must be understood and satisfied, or misleading results will be obtained. The authors examine these assumptions and qualitatively analyze the effect of these dangers when the FFT algorithm is supplied to power system load variation. They recommend the use of a DFT algorithm to evaluate the frequency spectrum of power system load variation. >
TL;DR: The Hartley Transform not only decreases the computer time of the WDF but also simplifies the convolution of two WDFs, which is used here to simulate a blurred image and its restoration.
NEC1
TL;DR: An improved FFT (fast Fourier transform) algorithm combining both decimations in frequency and in time is presented, and stress is placed on the derivation of general formulas for submatrices and multiplicands.
Abstract: An improved FFT (fast Fourier transform) algorithm combining both decimations in frequency and in time is presented. Stress is placed on a derivation of general formulas for submatrices and multiplicands. Computational efficiency is briefly discussed. >
TL;DR: A new fast discrete Radon transform method for enhancement of lines in noisy images is described, based on the Fourier slice theorem, with variable length slices to utilise all of the frequency domain data.
Abstract: A new fast discrete Radon transform method for enhancement of lines in noisy images is described. It is based on the Fourier slice theorem, with variable length slices to utilise all of the frequency domain data. It is shown that this new method achieves a significant increase in computational speed compared with an existing technique.
11 Apr 1988
TL;DR: A concept for a fixed point FFT (fast Fourier transform) error analysis is explained which allows a rather fast, simple and comprehensive numerical evaluation.
Abstract: A concept for a fixed point FFT (fast Fourier transform) error analysis is explained which allows a rather fast, simple and comprehensive numerical evaluation. This avoids time-consuming simulations or cumbersome theoretical derivations of the specific FFT length and structure for any change of the input signal or for any changes in the method of scaling and wordlength-reduction. From the results obtained, conclusions are drawn from reducing fixed-point errors with little or no additional effort. >
14 Nov 1988
TL;DR: In this article, the rough boundary is encoded in vector form by connecting the centroid of the region and its boundary points and resampling angularly, and the 2D boundary is transformed into a 1-D discrete curve.
Abstract: A scheme is proposed to estimate the boundary in complex imagery by the use of Fourier descriptors. The rough boundary is encoded in vector form by connecting the centroid of the region and its boundary points and resampling angularly. Thus the 2-D boundary is transformed into a 1-D discrete curve. The discrete Fourier transform (DFT) is applied to the transformed and resampled 1-D curve, and the Fourier descriptors are used to estimate the boundary of interest. Only a few of the harmonics (low-frequency components) are used for reconstruction of the boundary, and by adjusting the number of harmonics, a good estimation is obtained. The shrink-expand operation is incorporated with the Fourier descriptors to reduce irregularities parts on the boundary and to improve the accuracy of the resulting boundaries. >
11 Apr 1988
TL;DR: On taking the 2D Fourier transform of the output of a linear array, a ridge will appear on the frequency-wave number plane if a source is present and two methods are proposed to estimate this slope.
Abstract: On taking the 2D Fourier transform of the output of a linear array, a ridge will appear on the frequency-wave number plane if a source is present. The slope of this ridge is determined by the direction of arrival of the wavefront. Two methods are proposed to estimate this slope. The first one takes the sums of the squared magnitude of the transform along predetermined slopes to find the maximum sum. The second one requires less computation; it first locates all the maxima of the transform. A weighted-least-squares fit is then taken through these maxima to give a slope estimation. The multisource case is considered, and properties and statistics of the beamformer, together with simulation results, are given. >
TL;DR: According to this method, called the Running Fourier Transform (RFT) in this paper, running power spectra with arbitrary center frequencies and arbitrary Q values can be obtained.
Abstract: A short-time Fourier transform can be derived by low-pass filtering of the product of an input signal and exp(j2°ft). According to this method, called the Running Fourier Transform (RFT) in this paper, running power spectra with arbitrary center frequencies and arbitrary Q values can be obtained.
This paper proposes a fast algorithm of discrete RFT (FRFT). In the FRFT, a first-order lag system is adopted as the low-pass filter (LPF), and by approximating the impulse response of the LPF with a step function, the amount of multiplications is reduced. The calculation of the complex exponentials is omitted by referring to a table of sin(2°k/K) (k =0, …, K).