Showing papers on "Prime-factor FFT algorithm published in 1996"
15 Apr 1996
TL;DR: A new VLSI architecture for a real-time pipeline FFT processor is proposed, derived by integrating a twiddle factor decomposition technique in the divide-and-conquer approach, which has the same multiplicative complexity as the radix-4 algorithm, but retains the butterfly structure of the Radix-2 algorithm.
Abstract: A new VLSI architecture for a real-time pipeline FFT processor is proposed. A hardware-oriented radix-2/sup 2/ algorithm is derived by integrating a twiddle factor decomposition technique in the divide-and-conquer approach. The radix-2/sup 2/ algorithm has the same multiplicative complexity as the radix-4 algorithm, but retains the butterfly structure of the radix-2 algorithm. The single-path delay-feedback architecture is used to exploit the spatial regularity in the signal flow graph of the algorithm. For length-N DFT computation, the hardware requirement of the proposed architecture is minimal on both dominant components: log/sub 4/N-1 complexity multipliers and N-1 complexity data memory. The validity and efficiency of the architecture have been verified by simulation in the hardware description language VHDL.
410 citations
TL;DR: A multilevel algorithm is presented for analyzing scattering from electrically large surfaces that accelerates the iterative solution of integral equations that arise in computational electromagnetics.
Abstract: A multilevel algorithm is presented for analyzing scattering from electrically large surfaces. The algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics. The algorithm permits a fast matrix-vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks, with each describing the interaction between distant scatterers. The multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast Fourier transform (FFT) and that only relies on well-known algebraic techniques. The computational complexity and the memory requirements of the proposed algorithm are O(N log/sup 2/ N).
364 citations
01 Apr 1996
TL;DR: This paper surveys some recent work directed towards generalizing the fast Fourier transform (FFT) from the point of view of group representation theory, and discusses generalizations of the FFT to arbitrary finite groups and compact Lie groups.
Abstract: In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.
142 citations
TL;DR: A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fouriertransform (FFT) algorithm is presented and scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.
Abstract: A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT’s in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from −1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.
118 citations
TL;DR: A series-expansion approach and an operator framework are used to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction that is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small.
Abstract: We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N/sup 2/logN) floating-point operations per iteration to reconstruct an N/spl times/N image from P view angles, as compared to O(N/sup 2/P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.
97 citations
TL;DR: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest, but these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect.
Abstract: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest. Discrete Fourier transforms computed through the FFT are far more accurate than slow transforms, and convolutions computed via FFT are far more accurate than the direct results. However, these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect. Popular recursions for fast computation of the sine/cosine table (or twiddle factors) are inaccurate due to inherent instability. Some analyses of these recursions that have appeared heretofore in print, suggesting stability, are incorrect. Even in higher dimensions, the FFT is remarkably stable.
96 citations
TL;DR: Two programs are presented to compute direct- and cross-variogram values, direct andCross-covariograms, and pseudo-cross-variograms based on the Fast Fourier Transform algorithm, which is shown to be faster than the spatial approach for this type of data.
71 citations
01 Oct 1996-Precision Engineering-journal of The International Societies for Precision Engineering and Nanotechnology
TL;DR: A fast and reliable convolution algorithm to calculate the mean line of a roughness profile using the Gaussian filter according to ISO 11562 has been derived based on a recurrence relation for the weighting function.
Abstract: A fast and reliable convolution algorithm to calculate the mean line of a roughness profile using the Gaussian filter according to ISO 11562 has been derived. The algorithm is based on a recurrence relation for the weighting function. This greatly speeds up the calculation, making the algorithm nearly comparable to algorithms using the fast Fourier transform (FFT) in a usual way. The algorithm has been implemented as a short C function to be used with any evaluation program. The application of this function to a measured profile is given for demonstration and compared with the results obtained by the ordinary FFT filter algorithm.
55 citations
TL;DR: A fast algorithm, introduced by Brenier, which computes the Legendre-Fenchel transform of a real-valued function is investigated and the new approach of separating primal and dual spaces allows a clearer understanding of the algorithm and yields better numerical behavior.
Abstract: We investigate a fast algorithm, introduced by Brenier, which computes the Legendre-Fenchel transform of a real-valued function. We generalize his work to boxed domains and introduce a parameter in order to build an iterative algorithm. The new approach of separating primal and dual spaces allows a clearer understanding of the algorithm and yields better numerical behavior. We extend known complexity results and give new ones about the convergence of the algorithm.
49 citations
Patent•
26 Feb 1996
TL;DR: A real-time pipeline processor based on a hardware oriented radix-22 algorithm derived by integrating a twiddle factor decomposition technique in a divide and conquer approach is presented in this article.
Abstract: A real-time pipeline processor, which is particularly suited for VLSI implementation, is based on a hardware oriented radix-22 algorithm derived by integrating a twiddle factor decomposition technique in a divide and conquer approach. The radix-22 algorithm has the same multiplicative complexity as a radix-4 algorithm, but retains the butterfly structure of a radix-2 algorithm. A single-path delay-feedback architecture is used in order to exploit the spatial regularity in the signal flow graph of the algorithm. For a length-N DFT transform, the hardware requirements of the processor proposed by the present invention is minimal on both dominant components: Log4N-1 complex multipliers, and N-1 complex data memory.
46 citations
TL;DR: A simple algorithm is described for computing general pseudo-differential operator actions based on the asymptotic expansion of the symbol together with the fast Fourier transform, which shows that the algorithm is efficient through analyzing its complexity.
Abstract: A simple algorithm is described for computing general pseudo-differential operator actions. Our approach is based on the asymptotic expansion of the symbol together with the fast Fourier transform (FFT). The idea is motivated by the characterization of the pseudo-differential operator algebra. We show that the algorithm is efficient through analyzing its complexity. Some numerical experiments are also presented.
18 Jun 1996
TL;DR: It is found that for a certain dense set of fractional orders it is possible to define a discrete transformation and a fast algorithm is given, which has the same complexity as the FFT.
Abstract: Based on the fractional Fourier transformation of sampled periodic functions, the discrete form of the fractional Fourier transformation is obtained. It is found that for a certain dense set of fractional orders it is possible to define a discrete transformation. Also, for its efficient computation a fast algorithm, which has the same complexity as the FFT, is given.
TL;DR: It is shown that the EFT offers in a certain sense good time-frequency resolution and that stable reconstruction of a signal from samples of the E FT at equidistant time- frequencies grid points is possible, even for the case of nonredundant sampling.
13 Mar 1996
TL;DR: The idea of the method is to localize the nonregularities into the nodes of the Cooley-Tukey FFT type computational graph so that a simple programmable processor element for executing of node function can be the basis for parallel constructs.
Abstract: We propose for a class of trigonometric transforms fast algorithms with a unified structure and a simple data exchange similar to constant geometry isomorphic to the Cooley-Tukey FFT algorithm. One can easily extend many of the parallel FFT approaches for these algorithms. The idea of the method is to localize the nonregularities into the nodes of the Cooley-Tukey FFT type computational graph. Only the basic operation in the nodes of the computational graph will be different for different transforms. Thus a simple programmable processor element for executing of node function can be the basis for parallel constructs.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
07 May 1996
TL;DR: A fast algorithm for the computation of spherical harmonic expansions of bandlimited functions on the 2-sphere is described and it provides for an efficient inverse transform (synthesis) as well and consequently fast convolution on the2-spheres is also possible.
Abstract: A fast algorithm for the computation of spherical harmonic expansions of bandlimited functions on the 2-sphere is described. The algorithm also provides for an efficient inverse transform (synthesis) as well and consequently fast convolution on the 2-sphere is also possible. We discuss applications to image processing and medical imaging as well as aspects of our working implementation.
TL;DR: It is shown that while the technique presented herein is not expected to exhibit the same performance as that of comparable techniques based on the three-dimensional FFT, it is an attractive alternative that makes modest sacrifices in performance for gains in computational complexity.
Abstract: In this paper a description is given of a computationally efficient algorithm, based on the two-dimensional fast Fourier transform (FFT), for the estimation of multiple translational motions from a sequence of images. The proposed algorithm relies on properties of the projection (Radon) transform to reduce the problem from three to two dimensions and is effective in isolating and reliably estimating several superimposed motions in the presence of moderate levels of noise. Furthermore, the reliance of this algorithm on a novel array processing technique for line detection allows for the efficient estimation of the motion parameters. It is shown that while the technique presented herein is not expected to exhibit the same performance as that of comparable techniques based on the three-dimensional FFT, it is an attractive alternative that makes modest sacrifices in performance for gains in computational complexity.
TL;DR: A new parallel FFT algorithm is proposed that removes the complex multiplier between the two pipeline stages and simplifies the address generation of twiddle factors and reduces the number of twiddles to a minimum.
Abstract: Usually, parallel pipelined FFT processors are used to compute long FFTs due to high processing rate and easy implementation. The efficient VLSI implementation of each FFT processor at the pipelines is a critical problem to be considered. We propose a new parallel FFT algorithm that removes the complex multiplier between the two pipeline stages. The new algorithm also simplifies the address generation of twiddle factors and reduces the number of twiddle factors to a minimum. With the new algorithm, each FFT processor at the pipelines can be integrated easily onto a single chip.
TL;DR: A segmentation and feature based matching algorithm based on the Fourier- Mellin transform to rotation, translation and scaling and the use of the algorithm on two natural images is presented.
Abstract: Image matching methods should be invariant to translation, rotation and scale. This paper presents such an image matching algorithm based on Fourier-Mellin transform. An algorithm for calculation of Fourier-Mellin descriptors is first discussed. The invariance of Fourier-Mellin transform to rotation, translation and scaling is stated along with necessary proof. A segmentation and feature based matching algorithm based on the Fourier- Mellin transform is then presented. The use of the algorithm on two natural images is presented. The time complexity of the algorithm is also discussed.
TL;DR: In this paper, a discrete differentiation function is derived from the continuous derivative theorem of Fourier transforms, and the derivatives, semiderivatives and semi-integrals of theoretical and experimental voltammograms calculated using the algorithm are demonstrated.
TL;DR: The real and complex split-radix generalized fast Fourier transform algorithm has been developed and its applications for skew-circular convolution and partial FFT are described.
Patent•
27 Aug 1996
TL;DR: In this paper, the smallest possible circuit size is provided for FFT computing units, FFT computation devices, and pulse counters that can achieve computational precision using the smallest available circuit size.
Abstract: To provide FFT computing units, FFT computation devices, and pulse counters that can achieve computational precision using the smallest possible circuit size. FFT computing unit 602 comprises a data shift circuit for standardizing FFT computation target data to a specified bit width, adders/subtracters, multipliers, and data converters for standardizing the bit width to a certain bit width by truncating part of the output data of each computing unit, etc. FFT computation device comprises FFT computing unit 602, sensor 620, amplification circuit 621, gain control circuit 623, AD converter 622, first RAM 625 for sequentially storing the A/D conversion data, second RAM 626 for storing the FFT computation target data and the data being computed, coefficient ROM 101, and level determination circuit 624; and the level determination circuit determines the size of the data being transferred when the data is being transferred from RAM 1 to RAM 2, and the result is used for the data shift adjustment and gain control during FFT computation.
TL;DR: The empirical results for the Pearson X 2, likelihood ratio, and Freeman-Halton statistics show that the network algorithm, or equivalently, the recursive polynomial multiplication algorithm is superior to the FFT algorithm with respect to computing speed and accuracy.
05 May 1996
TL;DR: A fast algorithm for modified discrete cosine transform (MDCT), which is widely used in the coding of wide-band audio signals, is presented, which renders a simple and symmetric structure.
Abstract: This paper presents a fast algorithm for modified discrete cosine transform (MDCT), which is widely used in the coding of wide-band audio signals. By simple time shift and reverse, the MDCT is computed using Lee's (1984) fast cosine transform (FCT). The number of real multiplications is reduced from 2N/sup 2/ to (N/2)log/sub 2/N+N, half of that of the FFT based algorithm. The algorithm renders a simple and symmetric structure.
23 Oct 1996
TL;DR: It has been shown that the thresholding of the wavelet coefficients has near optimal noise reduction property for many classes of signals, and the proposed algorithm also reduces the noise while doing the approximation.
Abstract: We propose an algorithm that uses the discrete wavelet transform as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact results, and its computational complexity is on the same order of the FFT. The main advantage of the proposed algorithm is that the good time and frequency localization of wavelets can be exploited to approximate the Fourier transform for many classes of signals resulting in much less computation. Thus the new algorithm provides an efficient complexity vs accuracy tradeoff. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity. It has been shown that the thresholding of the wavelet coefficients has near optimal noise reduction property for many classes of signals. We show that for the same reason, the proposed algorithm also reduces the noise while doing the approximation. If we need to compute the DFT of noisy signals, the proposed algorithm not only can reduce the numerical complexity, but also can produce cleaner results. In summary, we propose a novel fast approximate Fourier transform algorithm using the wavelet transform. Since wavelets are the conditional basis of many classes of signals, the algorithm is very efficient and has built-in de-noising capacity.
TL;DR: A technique for partitioning hardware implementations of the Winograd (1976) Fourier transform algorithm (WFTA) into separate modules is presented, based on theWinograd nesting method, and thus preserves the minimum number of multiplications in the WFTA.
Abstract: A technique for partitioning hardware implementations of the Winograd (1976) Fourier transform algorithm (WFTA) into separate modules is presented. Instead of the prime factor algorithm, this technique is based on the Winograd nesting method and thus preserves the minimum number of multiplications in the WFTA. An integrated circuit capable of computing over 2 million 20-point discrete Fourier transforms/s is described. Using five of these integrated circuits, the partitioning technique can be applied to increase the transform length to 60 points.
07 May 1996
TL;DR: It is shown that detection performance increases monotonically with the number of FFT stages completed, converging ultimately to that of the exact ML detector.
Abstract: In the context of FFT-based maximum-likelihood (ML) detection of a complex sinusoid in noise, we consider the result of terminating the FFT at an intermediate stage of computation and applying the ML detection strategy to its unfinished results. We show that detection performance increases monotonically with the number of FFT stages completed, converging ultimately to that of the exact ML detector. The receiver operating characteristic associated with the completion of each FFT stage is derived. This enables the calculation of the minimum number of FFT stages that must be completed in order for desired detection and false alarm probabilities to be obtained.
14 Oct 1996
TL;DR: This work develops a real transform algorithm for calculating the discrete circular deconvolution by substituting the fast Fourier transform defined in the complex domain and it is shown that the computational cost is about half of the traditional FFT.
Abstract: Fast computation of the discrete deconvolution is very important in image/video signal processing. We develop a real transform algorithm for calculating the discrete circular deconvolution by substituting the fast Fourier transform (FFT) defined in the complex domain. It is shown that the computational cost of the algorithm is about half of the traditional FFT. Furthermore, the algorithm has a weak numerical stability.
TL;DR: In this paper, general systems of polynomials, satisfying prescribed symmetries and orthonormal on the unit circle with respect to weight functions belonging to a suitable symmetry class, are used in order to generalize the Discrete Fourier Transform (DFT) and the FFT algorithm.
Abstract: General systems of polynomials, satisfying prescribed symmetries and orthonormal on the unit circle with respect to weight functions belonging to a suitable symmetry class, are used in order to generalize the Discrete Fourier Transform (DFT) and the FFT algorithm.
TL;DR: Based on the differential property of Fourier transform and the Taylor expansion of a n-variables function, the subsequence interpolating algorithm is extended to a general n-dimensional signal as discussed by the authors.
Abstract: Based on the differential property of Fourier transform and the Taylor expansion of a n-variables function, the subsequence interpolating algorithm is extended to a general n-dimensional signal. As the interpolating process is consisted of a few parallel inverse FFT with the same size as the forward FFT, it is very efficient and is suitable for parallel processing.
27 May 1996
TL;DR: An efficient algorithm for the computation of 2D discrete Gabor transform under the assumption of non-overlapping windows, which is satisfied in many practical cases, the 2D Gabor coefficients can be calculated through an FFT-based scheme.
Abstract: An efficient algorithm for the computation of 2D discrete Gabor transform is introduced Under the assumption of non-overlapping windows, which is satisfied in many practical cases, the 2D Gabor coefficients can be calculated through an FFT-based scheme Combining with the Kohonen self-organized map, this method is used in a real-world problem dealing with cloud detection/classification Simulation results are also provided which show the promise of the proposed method