Showing papers on "Prime-factor FFT algorithm published in 1978"
TL;DR: In this paper, a new atomic-parameters least-squares refinement method is presented, which makes use of the fast Fourier transform algorithm at all stages of the computation.
Abstract: A new atomic-parameters least-squares refinement method is presented which makes use of the fast Fourier transform algorithm at all stages of the computation. For large structures, the amount of computation is almost proportional to the size of the structure making it very attractive for large biological structures such as proteins. In addition the method has a radius of convergence of approximately 0.75 A making it applicable at a very early stage of the structure-determination process. The method has been tested on hypothetical as well as real structures. The method has been used to refine the structure of insulin at 1.5 A resolution, barium beauvuricin complex at 1.2 A resolution, and myoglobin at 2 A resolution. Details of the method and brief summaries of its applications are presented in the paper.
185 citations
TL;DR: In this article, the authors show how discrete Fourier transformation can be implemented as a filter bank in a way which reduces the number of filter coefficients, leading to new forms of FFT's, among which is a \cos/sin FFT for a real signal which only employs real coefficients.
Abstract: The paper shows how discrete Fourier transformation can be implemented as a filter bank in a way which reduces the number of filter coefficients. A particular implementation of such a filter bank is directly related to the normal complex FFT algorithm. The principle developed further leads to types of DFT filter banks which utilize a minimum of complex coefficients. These implementations lead to new forms of FFT's, among which is a \cos/\sin FFT for a real signal which only employs real coefficients. The new FFT algorithms use only half as many real multiplications as does the classical FFT.
112 citations
TL;DR: This correspondence shows that if Haralick had made use of the fact that the FFT's of real sequences can be computed more rapidly than general FFTs, the result would have been reversed.
Abstract: Haralick has shown that the discrete cosine transform of N points can be computed more rapidly by taking two N-point fast Fourier transforms (FFT's) than by taking one 2N-point FFT as Ahmed had proposed. In this correspondence, we show that if Haralick had made use of the fact that the FFT's of real sequences can be computed more rapidly than general FFT's, the result would have been reversed. A modified algorithm is also presented.
77 citations
TL;DR: If the fast Fourier transfmm algorithm with n inputs, n as a power of two, is implemented with S temporary locations where S=o(n/ \log n) , then the computation time T grows faster than n\log n.
Abstract: The performance of the fast Fourier transfmm algorithm is examined under limitations on computational space and time. It is shown that if the algorithm with n inputs, n as a power of two, is implemented with S temporary locations where S=o(n/ \log n) , then the computation time T grows faster than n \log n . Furthermore, T can grow as fast as n^{2} if S=S_{min} + O(1) where S_{min}=l+\log_{2}n , the minimum necessary. These results are obtained by deriving tight bounds on T versus S and n .
60 citations
TL;DR: In an algorithm proposed here, DFT coefficients are computed via the Walsh transform (WT), which is superior to the fast Fourier transform (FFT) approach in applications where L is relatively small compared with N.
Abstract: This paper presents a new computational algorithm for the discrete Fourier transform (DFT). In an algorithm proposed here, DFT coefficients are computed via the Walsh transform (WT). The number of multiplications required by the new algorithm is approximately NL/6, where N is the number of data points and L is the number of Fourier coefficients desired. As such, it is superior to the fast Fourier transform (FFT) approach in applications where L is relatively small compared with N. It is also useful in cases where the Walsh and Fourier coefficients are both desired.
44 citations
TL;DR: The quantization error introduced by the Winograd Fourier transform algorithm (WFTA) when implemented in fixed-point arithmetic is studied and compared with that of the fast Fouriers transform (FFT).
Abstract: The quantization error introduced by the Winograd Fourier transform algorithm (WFTA) when implemented in fixed-point arithmetic is studied and compared with that of the fast Fourier transform (FFT). The effect of ordering the computational modules and the relative contributions of data quantization error and coefficient quantization error are determined. In addition, the quantization error introduced by the Good-Winogzad (GW) algorithm, which uses Good's prime-factor decomposition for the discrete Fourier transform (DFT) together with Winograd's short length DFT algorithms, is studied. Error introduced by the WFTA is, in all cases, worse than that of the FFT. In general, the WFTA requires one or two more bits for data representation to give an error similar to that of the FFT. Error introduced by the GW algorithm is approximately the same as that of the FFT.
29 citations
TL;DR: A calculational scheme is given for generating fluctuations which have any specified power spectrum and the fast computer-based algorithm makes use of a random number generator and fast Fourier transform routine.
Abstract: A calculational scheme is given for generating fluctuations which have any specified power spectrum. The fast computer-based algorithm makes use of a random number generator and fast Fourier transform (FFT) routine.
18 citations
TL;DR: The design for a convolution processor is presented, which employs a single highly parallel implementation of the fast Fourier transform (FFT) algorithm, eminently suited for real-time matched filtering of coded signals encountered in sonar systems.
Abstract: The design for a convolution processor is presented, which employs a single highly parallel implementation of the fast Fourier transform (FFT) algorithm. This processor is eminently suited for real-time matched filtering of coded signals encountered in sonar systems. Computer simulations have shown that this processor, which uses fixed point arithmetic and modest word sizes, can efficiently handle signals with multiple targets and relatively large Doppler shifts. The parallel architecture provides a throughput rate sufficient for computing both forward and inverse transforms in the one processor. The system is flexible permitting frequency domain adaptive beam-forming, attractive in many sonar applications.
10 citations
01 May 1978
TL;DR: It is shown that Winograd's algorithm can be used to compute an integer transform over GF(q), where q is a Mersenne prime, which makes it possible to more easily encode b.h.c. and r.s. codes.
Abstract: It is shown that Winograd's algorithm can be used to compute an integer transform over GF(q), where q is a Mersenne prime. This new algorithm requires fewer multiplications than the conventional fast Fourier transform (f.f.t). The transform over GF(q) can be implemented readily on a digital computer. This fact makes it possible to more easily encode b.c.h. and r.s. codes.
9 citations
TL;DR: This new hybrid algorithm requires fewer multiplications than any previously known algorithm and is a combination of a Winograd algorithm and a fast complex integer transform developed previously by the authors.
Abstract: In this paper it is shown that the cyclic convolution of complex values can be performed by a hybrid transform. This transform is a combination of a Winograd algorithm and a fast complex integer transform developed previously by the authors. This new hybrid algorithm requires fewer multiplications than any previously known algorithm.
9 citations
TL;DR: An algorithm which computes the Fourier transform of a sequence of length n over GF(2m) using approximately 2nm multiplications and n2+ nm additions is developed, which can be used when n is not highly composite or is a prime.
Abstract: An algorithm which computes the Fourier transform of a sequence of length n over GF(2m) using approximately 2nm multiplications and n2+ nm additions is developed. The number of multiplications is thus considerably smaller than the n2multiplications required for a direct evaluation, though the number of additions is slightly larger. Unlike the fast Fourier transform, this method does not depend on the factors of n and can be used when n is not highly composite or is a prime.
TL;DR: It is shown here that through a suitable ordering of calculations, the transforms over a complete set of overlapping "texture windows" can be obtained efficiently and be time-optimal to within a constant factor.
Abstract: The description of texture is an important problem in image analysis. Several methods in the literature require that local two-dimensional discrete Fourier transforms be computed as a first step in the texture description process. A chief limitation in these approaches has been the computational complexity of the transform calculation which has tended to limit the resolution of subsequent description and/or segmentation. It is shown here that through a suitable ordering of calculations, the transforms over a complete set of overlapping "texture windows" can be obtained efficiently. An algorithm is given and is shown to be time-optimal to within a constant factor.
TL;DR: This correspondence points out some inconsistencies between definitions and algorithms presented in the paper by H. F. Silverman.
Abstract: This correspondence points out some inconsistencies between definitions and algorithms presented in the paper by H. F. Silverman.
01 Apr 1978
TL;DR: WHT and RT appear to offer promise and potential compared to FFT as the former are easier to implement and as they yield recognition results comparable to those of the FFT.
Abstract: Traditionally FFT (fast implementation of discrete Fourier transform, DFT) has been utilized in recognition algorithms involving speech Other discrete orthogonal transforms such as Walsh-Hadamard transform (WHT) and rapid transform (RT) can play equally important roles in the recognition process as they have advantages in implementation and hardware realization The capability of these transforms in recognizing phonemes based on training matrices and mean square error (mse) criterion is investigated The speech data base consists of ten sentences spoken by ten different speakers (all male) For recognition purposes the speech is sectioned into 10 ms intervals and is sampled at 20 kHz Training matrices for all the three transforms are developed Test matrices in the transform domain are compared with the prototypes based on mse criterion which led to the decision process WHT and RT appear to offer promise and potential compared to FFT as the former are easier to implement and as they yield recognition results comparable to those of the FFT Other distance measures and recognition schemes are proposed for improving the classification accuracy
TL;DR: For certain large transform lengths, Winograd's algorithm for computing the discrete Fourier transform (d.f.t.) is extended considerably by performing the cyclic convolution, required by Winog Rad's method, by a fast transform over certain complex integer fields developed previously by the authors.
Abstract: For certain large transform lengths, Winograd's algorithm for computing the discrete Fourier transform (d.f.t.) is extended considerably. This is accomplished by performing the cyclic convolution, required by Winograd's method, by a fast transform over certain complex integer fields developed previously by the authors. This new algorithm requires fewer multiplications than either the standard fast Fourier transform (f.f.t.) or Winograd's more conventional algorithm.
IBM1
TL;DR: The purpose of this correspondence is to point out that in Table II of the above paper, the number of additions reported for the radix-2 FFT algorithm are highly erroneous.
Abstract: The purpose of this correspondence is to point out that in Table II of the above paper, the number of additions reported for the radix-2 FFT algorithm are highly erroneous.
IBM1
01 May 1978
TL;DR: Efficient algorithms for 11 and 13- point DFT's are presented and a more efficient algorithm, compared to earlier published versions, for the computation of 9-point DFT is also included.
Abstract: : Efficient algorithms for 11 and 13-point DFT's are presented. A more efficient algorithm, compared to earlier published versions, for the computation of 9-point DFT is also included. The effect of arithmetic roundoff in implementing the prime factor and the nested algorithms for computing DFT with fixed point arithmetic is analyzed using a statistical model. Various aspects of the prime factor, the nested and the radix-2 FFT algorithms are compared. A processor-based hardware implementation of the prime factor algorithm is discussed.
01 Jan 1978
TL;DR: An algorithm based on the Winograd method is developed to compute a Fourier-like transform over Galois field GF(2 exp n) for n equal to 5 and 6 and it is shown that this transform algorithm requires fewer multiplications than the more conventional fast transform algorithm described by Gentleman (1968).
Abstract: An algorithm based on the Winograd (1976) method is developed to compute a Fourier-like transform over Galois field GF(2 exp n) for n equal to 5 and 6. It is shown that this transform algorithm requires fewer multiplications than the more conventional fast transform algorithm described by Gentleman (1968). Such a transform can be used to encode and decode Reed-Solomon codes of length (2 exp n) -1.
01 Apr 1978
TL;DR: The concept of the zoom FFT (a more efficient algorithm which allows zooming in on a narrow segment of the spectrum while preserving its frequency content) is extended to the two-dimensional case and is further expanded to allow zooms over a specified segment within both the time and the frequency domains.
Abstract: The concept of the zoom FFT (a more efficient algorithm which allows zooming in on a narrow segment of the spectrum while preserving its frequency content) is extended to the two-dimensional case. The technique is further expanded to allow zooms over a specified segment within both the time and the frequency domains. Comparisons are also made as to the computational efficiency of this technique compared to the conventional two-dimensional FFT algorithms.
TL;DR: A new algorithm for computing efficiently multiplications by powers of a primitive element in the finite field GF(q2), where q is a Mersenne prime, is described, applicable to transforms over GF( q2) which is used to implement fast circular convolutions without roundoff error.
Abstract: In this correspondence a new algorithm for computing efficiently multiplications by powers of a primitive element in the finite field GF(q2), where q is a Mersenne prime, is described. This algorithm is applicable to transforms over GF(q2) which is used to implement fast circular convolutions without roundoff error.