Showing papers on "Cyclotomic fast Fourier transform published in 2002"

Studying Contact Stress Fields Caused by Surface Tractions With a Discrete Convolution and Fast Fourier Transform Algorithm

Abstract: The knowledge of contact stresses is critical to the design of a tribological element. It is necessary to keep improving contact models and develop efficient numerical methods for contact studies, particularly for the analysis involving coated bodies with rough surfaces. The fast Fourier Transform technique is likely to play an important role in contact analyses. It has been shown that the accuracy in an algorithm with the fast Fourier Transform is closely related to the convolution theorem employed. The algorithm of the discrete convolution and fast Fourier Transform, named the DC-FFT algorithm includes two routes of problem solving: DC-FFT/Influence coefficients/Green's, function for the cases with known Green's functions and DC-FFT/Influence coefficient/conversion, if frequency response functions are known. This paper explores the method for the accurate conversion for influence coefficients from frequency response functions, further improves the DC- FFT algorithm, and applies this algorithm to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy. A set of general formulas of the frequency response function for the elastic field is derived and verified. Application examples are presented and discussed.

The Cooley--Tukey FFT and Group Theory

Abstract: In 1965 J. Cooley and J. Tukey published an article detailing an efficient algorithm to compute the Discrete Fourier Transform, necessary for processing the newly available reams of digital time series produced by recently invented analog-to-digital converters. Since then, the Cooley– Tukey Fast Fourier Transform and its variants has been a staple of digital signal processing. Among the many casts of the algorithm, a natural one is as an efficient algorithm for computing the Fourier expansion of a function on a finite abelian group. In this paper we survey some of our recent work on he “separation of variables” approach to computing a Fourier transform on an arbitrary finite group. This is a natural generalization of the Cooley–Tukey algorithm. In addition we touch on extensions of this idea to compact and noncompact groups. Pure and Applied Mathematics: Two Sides of a Coin The Bulletin of the AMS for November 1979 had a paper by L. Auslander and R. Tolimieri [3] with the delightful title “Is computing with the Finite Fourier Transform pure or applied mathematics?” This rhetorical question was answered by showing that in fact, the finite Fourier transform, and the family of efficient algorithms used to compute it, the Fast Fourier Transform (FFT), a pillar of the world of digital signal processing, were of interest to both pure and applied mathematicians. Mathematics Subject Classification: 20C15; Secondary 65T10.

Low-complexity FFT structures for OFDM transceivers

Abstract: Orthogonal frequency-division multiplexing is a multiple-access technique with modulation and demodulation implemented by an inverse discrete Fourier transform (DFT) and a DFT, respectively. In a downlink (uplink) environment, an individual receiver (transmitter) may only use a small number of subchannels at any given time, in which case it does not make sense to require full DFT demodulation (inverse DFT modulation). Several existing low-complexity techniques for computing a partial DFT or inverse DFT with power-of-two size are examined. Low-complexity fast Fourier transform structures for full, few input, and few output nonpower-of-two transforms are derived.

A multiplier-less 1-D and 2-D fast Fourier transform-like transformation using sum-of-powers-of-two (SOPOT) coefficients

Abstract: This paper proposes a new multiplier-less approximation of the 1D discrete Fourier transform (DFT) called the multiplierless fast Fourier transform-like (ML-FFT) transformation. It parameterizes the twiddle factors in conventional radix-2/sup n/ or split-radix FFT algorithms as certain rotation-like matrices and approximates the associated parameters using the sum-of-powers-of-two (SOPOT) or canonical signed digits (CSD) representations. The ML-FFT converges to the DFT when the number of SOPOT terms used increases and has an arithmetic complexity of O(Nlog/sub 2/ N) additions, where N=2/sup m/ is the transform length. Design results show that the ML-FFT offers flexible tradeoff between arithmetic complexity and numerical accuracy in approximating the DFT. Using the polynomial transformation, similar multiplier-less approximation of 2D FFT is also obtained.

Spectral transformations for two-dimensional filters via FFT

Abstract: In this paper, a new fast algorithm for spectral transformations for two-dimensional digital filters is presented. The algorithm is based on the use of the fast Fourier transform. The computational complexity of this algorithm is evaluated. The simplicity and efficiency of the algorithm is illustrated by a numerical example.

Simultaneous DFT and IDFT of real N-point sequences

Abstract: The problem of simultaneously calculating the discrete Fourier transform (DFT) of a real N-point sequence and the inverse discrete Fourier transform (IDFT) of the DFT of a real N-point sequence using a single DFT is considered. New formulas are given that exploit the symmetries of the DFT.

Transfer Function Computation for Multidimensional Systems

Abstract: A theoretically interesting technique is proposed for the determination of the coefficients of the determinantal polynomial and the coefficients of the adjoint polynomial matrix of a given n-D system, described by the Fornasini-Marchesini state space model. The proposed algorithms are based on the discrete Fourier transform (DFT), and easily can be implemented. An example is included to illustrate the application of the algorithm.

WPT Based Fast Multiresolution Transform

Abstract: In this paper, we propose a fast multi-resolution transform using wavelet packet transform (WPT). This fast algorithm switches between a transform coder and a subband coder on user discretion. The proposed algorithm uses discrete approximate trigonometric expansions, which have previously been proposed for exploiting spatial and spectral correlation in multidimensional signals. Specifically, we describe an approach for fast implementation of the approximate Fourier expansion (AFE). This approach uses the discrete wavelet transform (DWT) as a tool to compute the approximate Fourier expansion (AFE). If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact result of the approximate Fourier expansion (AFE) of the signal, and its computational complexity is on the same order of the fast Fourier transform (FFT) algorithm. In this paper, we also show the capacity of the proposed algorithm for reducing noise while doing the approximation. Further, we discuss the possible implementation of the proposed algorithm using parallel processing resulting in faster implementation. The proposed algorithm provides an efficient complexity vs. accuracy tradeoff.

Iterative tomographic image reconstruction using Fourier-based forward and back-projectors

Abstract: Iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems, especially in emission tomography. With the fast increase of the sizes of the tomographic data, reduction of the computation demands of the reconstruction algorithms is of great importance. Fourier-based forward and back-projection methods have the potential to considerably reduce the computation time in iterative reconstruction. Additional substantial speed-tip of those approaches can be obtained utilizing powerful and cheap off-the-shelf FFT processing hardware. The Fourier reconstruction approaches are based on the relationship between the Fourier transform or the image and Fourier transformation of the parallel-ray projections. The critical two steps are the estimations of the samples of the projection transform, on the central section through the origin of Fourier space, from the samples of the transform of the image, and vice versa for back-projection. Interpolation errors are a limitation of Fourier-based reconstruction methods. We have applied min-max optimized Kaiser-Bessel interpolation within the nonuniform Fast Fourier transform (NUFFT) framework. This approach is particularly well suited to the geometries of PET scanners. Numerical and computer simulation results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic forward and back-projection than conventional interpolation methods, and that it is a viable candidate for fast iterative image reconstruction.

Approximation of Convolution Classes by Fourier Sums. New Results

Abstract: We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.

A MUL TlPLIER-LESS I-D AND 2-D FAST FOURIER TRANSFORM-LIKE TRANSFORMATION USING SUM-OF-POWERS-OF­

Abstract: This paper proposes a new mult iplier -less appro ximation of the I-D Discrete Fourier Transform (DFD called the multiplier­ less Fast Fourier Transform-like (ML-FFD transformation. It para meteri zes the twiddle factors in conventional radix- 2" or split-radix FFT algorithms as certain rotation-like matrices and approximates the associated parameters the sum-of­ powers-of-two (SOPOD or canonical signed digits (CSD) representations. The ML-FFT converges to the OFT when the number of SOPOT terms used increases and has an arithmetic complexity of O(Nlog, N) addi tions, where N = 2m is the transform length. Oesign results show that the ML-FFT offers flexible tradeoff between arithmetic complexity and numerical accuracy in approximating the DFT. Using the polyno mial transformation, similar multiplier-less approximation of 2-D FFT is also obtain ed.