Showing papers on "Prime-factor FFT algorithm published in 1977"
TL;DR: A Fast Discrete Cosine Transform algorithm has been developed which provides a factor of six improvement in computational complexity when compared to conventional DiscreteCosine Transform algorithms using the Fast Fourier Transform.
Abstract: A Fast Discrete Cosine Transform algorithm has been developed which provides a factor of six improvement in computational complexity when compared to conventional Discrete Cosine Transform algorithms using the Fast Fourier Transform. The algorithm is derived in the form of matrices and illustrated by a signal-flow graph, which may be readily translated to hardware or software implementations.
1,301 citations
TL;DR: Two recently developed ideas, the conversion of a discrete Fourier transform to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms.
Abstract: Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented
331 citations
IBM1
TL;DR: It is shown how the Chinese Remainder Theorem can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions and can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm.
Abstract: It is shown how the Chinese Remainder Theorem (CRT) can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions. Then, special algorithms are developed which, compute the relatively short convolutions in each of the dimensions. The original suggestion for this procedure was made in order to extend the lengths of the convolutions which one can compute with number-theoretic transforms. However, it is shown that the method can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm. Some of the short convolutions are computed by methods in an earlier paper by Agarwal and Burrus. Recent work of Winograd, consisting of theorems giving the minimum possible numbers of multiplications and methods for achieving them, are applied to these short convolutions.
257 citations
IBM1
TL;DR: A new approach to the computation of the discrete Fourier transform (DFT) with significantly reduced number of multiplication operations; it does not increase the number of addition operations in many cases.
Abstract: Recently, Dr. Shmuel Winograd discovered a new approach to the computation of the discrete Fourier transform (DFT). Relative to fast Fourier transform (FFT), the Winograd Fourier transform algorithm (WFTA) significantly reduces the number of multiplication operations; it does not increase the number of addition operations in many cases. This paper introduces the new algorithm and discusses the operations comparison problem. A guide for programming is included, as are some preliminary running times.
178 citations
TL;DR: In this article, a method for calculating structure factors by Fourier inversion of a model electron density map is presented, which is 3½ to 7 times less expensive than conventional methods for non-centrosymmetric space groups.
Abstract: A method is presented for calculating structure factors by Fourier inversion of a model electron density map. The cost of this method and of the standard methods are analyzed as a function of number of atoms, resolution, and complexity of space group. The cost functions were scaled together by timing both methods on the same problem, with the same computer. The FFT method is 3½ to 7 times less expensive than conventional methods for non-centrosymmetric space groups.
105 citations
TL;DR: A generalized mathematical theory of holor algebra is used to manipulate coefficient arrays needed to generate computational equations which involve elements from throughout the two-dimensional array rather than operating on individual rows and columns.
Abstract: A mathematical development is presented for a direct computation of a two-dimensional fast Fourier transform (FFT). A generalized mathematical theory of holor algebra is used to manipulate coefficient arrays needed to generate computational equations. The result is a set of equations which involve elements from throughout the two-dimensional array rather than operating on individual rows and columns. Preliminary digital computer calculations verify the accuracy of the technique and demonstrate a modest saving of computation time as well.
74 citations
01 May 1977
TL;DR: In this paper, the authors simplify the concepts of the zoom transform and remove some of the restrictions assumed by Yip; i.e., the total number of points need not be a power of 2.
Abstract: A recent paper by Yip discussed the zoom transform as derived from the defining equation of the FFT. This paper simplifies the concepts and removes some of the restrictions assumed by Yip; ie., the total number of points need not be a power of 2. The technique is based on first specifying the desired center frequency, bandwidth, and frequency resolution. The signal is then sampled, modulated, and lowpass filtered. This result is purposely aliased, then transformed using an FFT algorithm. The result is an M-point frequency spectra of the desired bandwidth centered about the center frequency with a higher degree of resolution than could be directly obtained using an M-point transform.
49 citations
TL;DR: The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT and is used to filter a two-dimensional picture, and the results are presented with a comparison to the standard FFT.
Abstract: A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.
44 citations
01 Jul 1977
TL;DR: In this article, a 1D algorithm using the Hankel transform of the section of the function is described, which can avoid the use of the 2D FFT algorithm due to the loss of symmetry due to sampling and to a waste in storage requirements.
Abstract: Computing the Fourier transform of a circularly symmetric function is often necessary in optics. Use of the 2-D FFT algorithm leads to loss of the symmetry because of the sampling and to a waste in storage requirements; to avoid these inconveniences, a 1-D algorithm is described using the Hankel transform of the section of the function.
20 citations
TL;DR: The results show that the error performance of the decimation-in-frequency algorithm is better than that of decimation -in-time, and two kinds of schemes for preventing overflow are considered in the analysis.
Abstract: This correspondence presents some results in fixed-point error analysis of fast Fourier transform algorithms. Two kinds of schemes for preventing overflow are considered in the analysis. The results, obtained for the decimation-in-frequency form of the algorithm, are compared with those of decimation-in-time. The results show that the error performance of the decimation-in-frequency algorithm is better than that of decimation-in-time.
20 citations
01 May 1977
TL;DR: This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process, which is extremely valuable on small disc based computers.
Abstract: Conventional two dimensional fast Fourier transforms become very slow if the size of the matrix becomes too large to be contained in memory. This is due to the transposition of the matrix that is required. This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process. This algorithm is extremely valuable on small disc based computers.
IBM1
TL;DR: One "General-N" (i.e. many allowable DFT sizes (N) but certainly not any vector size) complex WFTA programming technique is described.
Abstract: The Winograd Fourier Transform Algorithm (WFTA) requires about 20% of the multiplications used in an optimized FFT, while the number of additions remains unchanged. This paper describes one "General-N" (i.e. many allowable DFT sizes (N) but certainly not any vector size) complex WFTA programming technique.
TL;DR: A statistical model for roundoff error is used to predict the output noise to signal ratio of the two common FFT algorithms, the decimation in time and the decimating in frequency algorithms.
Abstract: A statistical model for roundoff error is used to predict the output noise to signal ratio of the two common FFT algorithms, the decimation in time and the decimation in frequency algorithms. A unified approach is used to obtain the error in both algorithms. Results for radix 2 and for arbitrary radix are presented. Multidimensional FFT is also discussed.
TL;DR: In this paper, a formula for interpolation between output samples of a fast Fourier transform (FFT) is derived for obtaining greater frequency resolution when two coarse FFT outputs are available.
Abstract: A formula is derived for interpolation between output samples of a fast Fourier transform (FFT), i.e., in the frequency domain. Such a formula is useful for obtaining greater frequency resolution when two coarse FFT outputs are available. Consideration is also given to the effect of such interpolation on a weighted FFT.
TL;DR: This work shows how to perform a number-theoretic transform (n.t.t.) using an algorithm analogous to that of S.s. Winograd for computing the discrete Fourier transform (d.f.t).
Abstract: We show how to perform a number-theoretic transform (n.t.t.) using an algorithm analogous to that of S. Winograd for computing the discrete Fourier transform (d.f.t.). Using this algorithm, the range of data lengths and word lengths is much larger than that available with conventional fast n.t.t.s.
TL;DR: In this correspondence both high-radix and real-valued input FFT algorithms are applied to transforms over the finite field GF(q2), where q is a Mersenne prime, which can be used to implement fast circular convolutions without roundoff error.
Abstract: In this correspondence both high-radix and real-valued input FFT algorithms are applied to transforms over the finite field GF(q2), where q is a Mersenne prime. Such transforms can be used to implement fast circular convolutions without roundoff error. Of particular interest is a new radix 8 FFT algorithm, which requires fewer multiplications than the conventional radix 8 FFT algorithm.
TL;DR: In this paper, it was shown that a high-radix fast Fourier transform (FFT) with generator \gamma = 3 over GF (F n) can be used for encoding and decoding of Reed-Solomon (RS) codes of length 2^{2}n}.
Abstract: It is shown that a high-radix fast Fourier transform (FFT) with generator \gamma = 3 over GF (F_{n}) , where F_{n} = 2^{2}^{n'} + 1 is a Fermat prime, can be used for encoding and decoding of Reed-Solomon (RS) codes of length 2^{2}^{n} . Such an RS decoder is considerably faster than a decoder using the usual radix 2 FFT. This technique applies most ideally to a 16-error-correcting, 256-symbol RS code of 8 bits being considered currently for space communication applications. This special code can be encoded and decoded rapidly using a high-radix FFT algorithm over GF (F_{3}) .
15 Jun 1977
TL;DR: A high-radix fast Fourier transformation (FFT) algorithm for computing transforms over GF(sq q), where q is a Mersenne prime, is developed to implement fast circular convolutions.
Abstract: A high-radix fast Fourier transformation (FFT) algorithm for computing transforms over GF(sq q), where q is a Mersenne prime, is developed to implement fast circular convolutions. This new algorithm requires substantially fewer multiplications than the conventional FFT.
01 Jan 1977
TL;DR: A discrete Fourier transform module for incorpration in fast Fourier Transform processors is described, which is highly suitable for real input applications requiring high-speed transformations.
Abstract: For applications requiring high-speed and in-place treatment, it is often advantageous to realize special-purpose computers. This paper describes a discrete Fourier transform (DFT) module for incorpration in fast Fourier transform (FFT) processors. The module is highly suitable for real input applications requiring high-speed transformations. It attributes one point to all frequency channels in one clock cycle. This treatment is not only well suited for the present technology, but appears to be more attractive in view of recent trends in digital circuitry.
Proceedings Article•
22 Aug 1977TL;DR: This paper presents an efficient method to calculate two-dimensional discrete Fourier transforms over windowed regions of the light intensity matrix based on the fast Fourier transform algorithm, which can be beaten by any nonparallel algorithm.
Abstract: Computer vision systems based on general purpose computers often need efficient texture description algorithms. One common method is to calculate two-dimensional discrete Fourier transforms over windowed regions of the light intensity matrix. Although these methods described in the literature are based on the fast Fourier transform algorithm, the computation time is still too high to permit the description of texture for as many windows as are needed for good segmentation. When a set of transforms over a window at every position of the matrix is needed, an efficient method can be used. It saves information computed for previous windows and uses it to reduce the effort expended on the current window. For a window N × N and an image matrix M × M, the time complexity is reduced from O(N2M2logN) to O(N2M2). This complexity cannot be beaten by any nonparallel algorithm.
TL;DR: This paper shows how a combination of the techniques of “redundancies” and “Kronecker decompositions” may be used with a specification for the data collection to produce a fast, accurate algorithm which solves the linear system.
Abstract: In a recently developed approach to the optical inverse scattering problem, the need arose to solve very large systems of linear equations with a nonsparse matrix. The entries in the matrix are determined by the specifications for the data collection pattern. The matrix is not a Discrete Fourier Transform matrix, and it not anenable to FFT methods. In this paper we show how a combination of the techniques of “redundancies” and “Kronecker decompositions” may be used with a specification for the data collection to produce a fast, accurate algorithm which solves the linear system. This algorithm has been implemented on a sequential computer, but parallel computation is clearly feasible.
01 May 1977
TL;DR: In an algorithm proposed here, first the discrete Walsh transform of sampled data is evaluated and then using these results Fourier coefficients can be computed, which is useful in the case where the number of L is not very large, or Walsh coefficients and Fourier coefficient are both calculated.
Abstract: This paper presents a new computational algorithm for the discrete Fourier transform. In an algorithm proposed here, first the discrete Walsh transform of sampled data is evaluated and then using these results Fourier coefficients can be computed. The number of multiplications in the algorithm can be expressed by approximately NL/6 for N data points and L Fourier coefficients to be calculated. On the other hand, the fast Fourier transform must compute all of Fourier coefficients independently of L. This algorithm is useful in the case where the number of L is not very large, or Walsh coefficients and Fourier coefficients are both calculated.