Showing papers on "Prime-factor FFT algorithm published in 1982"
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TL;DR: The Fast Fourier Transform (FFT) algorithm as discussed by the authors is the most well known vectorization algorithm for the discrete complex Fourier transform (DCFT). But it is not a generalization of the slow transform.
Abstract: Publisher Summary This chapter provides an overview on vectorizing the FFTs. The fast Fourier transform (FFT) is the most well known of all algorithms. It is superior to the slow transform and has applications in all areas of scientific computing. The term FFT was applied to a specific algorithm for the rapid computation of the discrete complex Fourier transform; however, it has become a generic term that is applied to any one of a large number of algorithms that compute the complex as well as other Fourier transforms. Many algorithms exist for a given Fourier transform, and when they are applied to a particular sequence, the result is the same. However, the algorithms differ in the ways that intermediate results are computed and stored. It is these important differences that provide the algorithms with unique properties that make one or the other more attractive for a particular application.
151 citations
TL;DR: An algorithm that reduces the computational effort to two-thirds of the effort required by most radix-2 algorithms and its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences.
Abstract: This paper develops and presents a radix-2 fast Fourier transform (FFT) algorithm that reduces the computational effort (as measured by the number of multiplications) to two-thirds of the effort required by most radix-2 algorithms. The resulting algorithm is similar to one obtained by applying a principle suggested by Rader and Brenner; however, its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences; furthermore, memory requirements (other than those for storing the input data) do not grow with the size of the transform.
43 citations
TL;DR: A modified version of Burrus' prime factor fast Fourier transform program is described, which implements the in-place, in-order algorithm for variable transform sizes.
Abstract: This paper describes a modified version of Burrus' prime factor fast Fourier transform program. The modifications produce a general-purpose program which implements the in-place, in-order algorithm for variable transform sizes. Speed tests show the resulting program to be faster than a program using a separate reordering pass.
34 citations
TL;DR: A Fast Fourier Transform algorithm is described which is especially suited for structural dynamics which incorporates several features selected from many variations of the original Cooley and Tukey1 algorithm with the goal of making the most efficient use of computer time and storage while maintaining simplicity.
Abstract: A Fast Fourier Transform algorithm (FFT) is described which is especially suited for structural dynamics. The routine incorporates several features selected from many variations of the original Cooley and Tukey1 algorithm with the goal of making the most efficient use of computer time and storage while maintaining simplicity. Some introductory material to Fourier transform techniques and a description of the original algorithm are also included. In addition, the source listing of the subroutine FFT is reproduced.
29 citations
TL;DR: The eigensystem for the Fast Fourier transform, FFT, known for several years, can be used to design FFT algorithms and it is found that for every prime number transform there are only 4 distinct eigenvalues.
20 citations
TL;DR: A time-efficient algorithm for calculating the discrete Fourier transform is developed which uses a prime factor decomposition of the DFT into multiple short prime length DFT's which are converted into cyclic convolutions by an index permutation based on number theory.
Abstract: A time-efficient algorithm for calculating the discrete Fourier transform is developed. It uses a prime factor decomposition of the DFT into multiple short prime length DFT's which are converted into cyclic convolutions by an index permutation based on number theory. The convolutions are evaluated by table look-up using distributed arithmetic. When programmed on a Z80 microprocessor, the algorithm is 2-20 times faster than conventional algorithms. The approach also makes it possible to add simple external logic to a micro-processor system to further increase the speed.
18 citations
TL;DR: A new algorithm for the calculation of the Fourier transform of sampled time functions is described, based on second‐degree polynomial interpolations between the sample points, which was found to be significantly more accurate than the conventionally used discrete Fouriertransform (DFT).
Abstract: A new algorithm for the calculation of the Fourier transform of sampled time functions is described. The algorithm is especially applicable to the Fourier analysis of nonperiodic signals which are not band limited. The method is based on second‐degree polynomial interpolations between the sample points. The obtained continuous approximation of the signal allows the determination of the Fourier transform analytically. In the case of exponentially decaying functions the algorithm was found to be significantly more accurate than the conventionally used discrete Fourier transform (DFT). The computing time is only about twice the time required by the fast Fourier transform (FFT) algorithm.
16 citations
03 May 1982
TL;DR: A highly effective dynamic programming algorithm is presented as a solution to the problem of finding an algorithm from this class which is optimal with respect to the specific add, multiply, and data transfer characteristics of a partlcular implementation.
Abstract: A broad class of efficient, discrete Fourier transform algorithms is developed by partitioning short DFT algorithms into factors. The factored short DFT's are combined into longer DFT's using a prime factor algorithm (PFA). By exploiting a property which allows some of the factors to commute, a large set of possible DFT algorithms is generated which contains both the prime factor algorithm and the Winograd Fourier Transform Algorithm (WFTA) as special cases. The problem of finding an algorithm from this class which is optimal with respect to the specific add, multiply, and data transfer characteristics of a partlcular implementation is posed, and a highly effective dynamic programming algorithm is presented as a solution.
15 citations
TL;DR: It is shown that Singleton's mixed radix algorithm (MFFT) is the most flexible and uses the least memory, while the Winograd Fourier transform algorithm (WFTA) and Kolba-Parks prime factor algorithm (PFA) are the most efficient.
Abstract: The number of real operations and memory is presented for three efficient Fortran algorithms which compute the mixed radix discrete Fourier transform (DFT). It is shown that Singleton's mixed radix algorithm (MFFT) is the most flexible and uses the least memory, while the Winograd Fourier transform algorithm (WFTA) and Kolba-Parks prime factor algorithm (PFA) are the most efficient.
11 citations
TL;DR: This paper reports on the application of a new FFT algorithm, first described by Winograd, to the calculation of diffraction OTF that yields the same accuracy as that obtained by the Cooley-Tukey method but is up to four times faster.
Abstract: Although fast Fourier transform (FFT) algorithms based on the Cooley-Tukey method have been widely used for the computation of optical transfer function (OTF), the need for yet faster algorithms remains. This is particularly so since desk-top computers with modest speed and memory size have become essential tools in optical design. In this paper we report on the application of a new FFT algorithm, first described by Winograd, to the calculation of diffraction OTF. The algorithm is compared both in speed and in accuracy with the commonly used radix-2 FFT and with an autocorrelation method employing the Gaussian quadrature integration technique. It is found that the new algorithm yields the same accuracy as that obtained by the Cooley-Tukey method but is up to four times faster. Some other advantages and drawbacks are discussed.
6 citations
TL;DR: An FFT algorithm operating on a 16-bit microcomputer can calculate a 256-point transform as much as 10 times faster than a similar algorithm on an eight- bit microcomputer.
Abstract: An FFT algorithm operating on a 16-bit microcomputer can calculate a 256-point transform as much as 10 times faster than a similar algorithm on an eight-bit microcomputer.
07 Jun 1982
TL;DR: This paper analyzes in detail how far the proposed Single Instruction Multiple Data computers with interconnection networks are applicable in the signal processing area and shows that the latter gives much higher speedup for P > 16 and is more cost-effective even with the high cost of switches.
Abstract: This paper analyzes in detail how far the proposed Single Instruction Multiple Data (SIMD) computers with interconnection networks are applicable in the signal processing area. Decimation in the time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in a multiprocessor system with shared bus and an SIMD computer with interconnection network.Results are derived for data allocation, interprocessor communication, approximate computation time, speedup, and cost effectiveness for an N-point FFT with any P available processors. Further generalization is obtained for a radix-r FFT algorithm. N X N point, two-dimensional discrete Fourier transform (DFT) implementation is also considered, with one or more rows of input matrix allocated to each processor.Various curves are plotted and a comparison in performance is carried out between a shared-bus multiprocessor and SIMD computer with interconnection network. It is shown that the latter gives much higher speedup for P > 16 and is more cost-effective even with the high cost of switches. N, P and r, considered here, are all powers of 2.
TL;DR: New recursive formulae for trigonometrischen functions generation suitable for FFT algorithms are given and Evaluation of one pair of trigonometric coefficients thus requires 2 multiplications and 2 additions only.
Abstract: New recursive formulae for trigonometric functions generation suitable for FFT algorithms are given. Evaluation of one pair of trigonometric coefficients thus requires 2 multiplications and 2 additions only. Speed comparisons of various radices 2, 4 and 8 FFT FORTRAN realizations were performed. An efficient FORTRAN IV program for one-dimensional complex FFT based on radix 8 algorithm with recursively generated trigonometric coefficients is supplied.
03 May 1982
TL;DR: Two algorithms for the calculation of Fourier transform of a discrete signal are derived from the known recursive method for polynomial evaluation and the relation of these algorithms to the well known Goertzel algorithm and the Collatz's rule is demonstrated.
Abstract: Two algorithms for the calculation of Fourier transform of a discrete signal are derived from the known recursive method for polynomial evaluation. The first algorithm processes the elements of the discrete signal in a natural order of elements and the second one in the reverse order. Both algorithms are modified for operation with real numbers only. The relation of these algorithms to the well known Goertzel algorithm and the Collatz's rule is demonstrated. Moreover, the application of the recursive algorithm to repeated DFT calculation is described.
01 May 1982
TL;DR: A new two dimensional DFT algorithm is given, the Mixed Simultaneous Decimation FFT algorithm, which can be used to perform decimation-in-time orDecimation-infrequency in both dimensions simultaneously.
Abstract: The well-known decimation-in-time and decimation-infrequency FFT algorithms have recently been combined into a single and more efficient one, the Mixed Decimation FFT algorithm. On the other hand, an efficient way to perform a two dimensional DFT computation is to use decimation-in-time or decimation-infrequency in both dimensions simultaneously. In this paper the above two ideas are combined to give a new two dimensional DFT algorithm, the Mixed Simultaneous Decimation FFT algorithm.
IBM1
TL;DR: Comparisons demonstrate a two to four times throughput improvement for the use of the polynomial transform method in place of the 2D FFT approach to circularly convolve or generate 2D Fourier transforms for large 2D fields in the range 1024 × 1024 to 8192 × 8192.
Abstract: This paper presents new results concerning the matrix data re-order requirements of polynomial-transform-based 2D convolution and 2D Fourier Transform methods which can be employed in digital processing of images and other 2D problems. The results indicate that several power-of-2 length-modified ring polynomial transform methods developed by Nussbaumer allow the total avoidance of the row-column matrix transpose commonly encountered in other algorithmic approaches, while also providing a number of other computational advantages. It is demonstrated that this property can be the source of significantly improved throughput on a number of existing data processing structures. An execution time comparison with an efficient Fast Fourier Transform algorithm base is made assuming the use of general register architecture and array processor units. It is also assumed that one makes use of recently developed efficient matrix transpose methods by Eklundh and Ari to support 2D FFT data re-order requirements. These comparisons demonstrate a two to four times throughput improvement for the use of the polynomial transform method in place of the 2D FFT approach to circularly convolve or generate 2D Fourier transforms for large 2D fields in the range 1024 × 1024 to 8192 × 8192.
01 May 1982
TL;DR: The statistical model is assumed to determine errors in the computatlon of discrete Fourier transform with the use of mixed-radix FFT algorithm with decimation in frequency and the formulae for calcu1ati.
Abstract: The statistical model is assumed to determine errors a r i s i n g @ the computatlon of discrete Fourier transform with the use of mixed-radix FFT algorithm with decimation in frequency. The fixed point arithmetic operations on the numbers represented in two's complenunt code are considered. The formulae for calcu1ati.g the ,mean s q a r e value of errors in sny point of transform outpot for 'the operations of rounding with andwithout scaling, and chopping with and without scaling are given in the paper. Computer. simulation was carried out and general conclusions concerning mixed- -radix FFT algorithms with decimatiou in frequency were drawn.
01 Nov 1982
TL;DR: The notation is in this paper used to derive a description of array implementations of decimation-in-frequency and decimation -in-time FFT algorithms.
Abstract: Fast Fourier Transform, FFT, algorithms are interesting for direct hardware implementation in VLSI. The description of FFT algorithms is typically made either in terms of graphs illustrating the dependency between different data elements or in terms of mathematical expressions without any notion of how the computations are implemented in space or
time. Expressions in the notation used in this paper can be given an interpretation in the implementation domain. The notation is in this paper used to derive a description of array implementations of decimation-in-frequency and decimation-in-time FFT algorithms. Correctness of the implementations is guaranteed by way of derivation.
28 Dec 1982
TL;DR: A new high capacity, high speed FFT architecture is presented which is being incorporated in a nearly completed development model for a flexible series of FFT processors.
Abstract: Fast Fourier transform methods are attractive techniques for implementing high speed programmable filters, especially when flexibility, accuracy, and sharp filter transition regions are important considerations. In this paper are some design considerations in the implementation of such filters. A new high capacity, high speed FFT architecture is presented which is being incorporated in a nearly completed development model for a flexible series of FFT processors. It is shown how this processor may be efficiently embedded in an overall programmable filtering structure.© (1982) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: The effect of pitch period length and position with respect to the analysis frame is described, which may introduce distortions in spectra estimation in both pitch-synchronous and pitch-asynchronous analysis.