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Showing papers on "Prime-factor FFT algorithm published in 1979"


Journal ArticleDOI
TL;DR: In this article, interpolated fast Fourier transform (FFT) algorithms are used for multi-parameter measurements upon periodic signals, such as fundamental frequency, phase, and amplitude, with enhanced accuracy compared to existing algorithms.
Abstract: By use of an interpolated fast-Fourier-transform (FFT) algorithms are developed for multiparameter measurements upon periodic signals. Eight pertinent measurements, such as fundamental frequency, phase, and amplitude, are made with enhanced accuracy compared to existing algorithms, including tapered-window-FFT algorithms. For the more general case of nonharmonic multitone signals also the method is shown to yield exact amplitudes and phases if the tone frequencies are known beforehand. These measurements are useful in a variety of applications ranging from analog testing of printed-circuit boards to measurement of Doppler signals in radar detection.

421 citations


Journal ArticleDOI
TL;DR: A new method of deriving very fast Fourier transform algorithms that do not employ multiplication and have a form suitable for high performance hardware implementations is described.
Abstract: A new method of deriving very fast Fourier transform (FFT) algorithms is described. The resulting algorithms do not employ multiplication and have a form suitable for high performance hardware implementations. The complexity of the algorithms compares favorably to the recent results of Winograd [1].

126 citations


Journal ArticleDOI
Henri J. Nussbaumer1, P. Quandalle1
TL;DR: In this article, two polynomial transforms have been proposed for computing discrete Fourier transform (DFT) by polynomials, which are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFTs.
Abstract: Polynomial transforms, defined in rings of polynomials, have been introduced recently and have been shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFT's and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA). We also describe new split nesting and split prime factor techniques for computing large DFT's from a small set of short DFT's with a minimum number of operations.

63 citations


Journal ArticleDOI
TL;DR: A statistical analysis of the transform domain displacement estimation algorithm and its convergence under certain realistic conditions is given and an extension of the algorithm that adaptively updates displacement estimation according to the local features of the moving objects is described.
Abstract: This paper introduces an algorithm for estimating the displacement of moving objects in a television scene from spatial transform coefficients of successive frames. The algorithm works recursively in such a way that the displacement estimates are updated from coefficient to coefficient. A promising application of this algorithm is in motion-compensated interframe hybrid transform- dpcm image coding. We give a statistical analysis of the transform domain displacement estimation algorithm and prove its convergence under certain realistic conditions. An analytical derivation is presented that gives sufficient conditions for the rate of convergence of the algorithm to be independent of the transform type. This result is supported by a number of simulation examples using Hadamard, Haar, and Slant transforms. We also describe an extension of the algorithm that adaptively updates displacement estimation according to the local features of the moving objects. Simulation results demonstrate that the adaptive displacement estimation algorithm has good convergence properties in estimating displacement even for very noisy images.

37 citations


Journal ArticleDOI
TL;DR: The algorithm uses the fast Fourier transform to diagonalize and decouple the system of equations which results from the application of the least-squares criterion and is accurate and stable, and is perhaps an order of magnitude faster than the best iterative method.
Abstract: This paper describes a fast direct algorithm for obtaining least-squares phase estimates from arrays of noisy phase differences. The algorithm uses the fast Fourier transform to diagonalize and decouple the system of equations which results from the application of the least-squares criterion. It is accurate and stable, and is perhaps an order of magnitude faster than the best iterative method. The effectiveness of the algorithm has been demonstrated by using it in connection with the Knox–Thompson speckle-imaging procedure to restore an optical object perturbed by simulated atmospheric turbulence. Representative results are discussed in the paper.

29 citations


Journal ArticleDOI
Corsini1, Frosini
TL;DR: In this work the generalized discrete Fourier transform (GFT), which includes the DFT as a particular case, is considered, and two pairs of fast algorithms for evaluating a multidimensional GFT are given (T-algorithm, F-al algorithm, and T′-algorithms, F′-Algorithm).
Abstract: In this work the generalized discrete Fourier transform (GFT), which includes the DFT as a particular case, is considered. Two pairs of fast algorithms for evaluating a multidimensional GFT are given (T-algorithm, F-algorithm, and T′-algorithm, F′-algorithm). It is shown that in the case of the DFT of a vector, the T-algorithm represents a form of the classical FFT algorithm based on a decimation in time, and the F-algorithm represents a form of the classical FFT algorithm based on decimation in frequency. Moreover, it is shown that the T′-algorithm and the T-algorithm involve exactly the same arithmetic operations on the same data. The same property holds for the F′-algorithm and the F-algorithm. The relevance of such algorithms is discussed, and it is shown that the T′-algorithm and the F′-algorithm are particularly advantageous for evaluating the DFT of large sets of data.

28 citations


Journal ArticleDOI
TL;DR: An application to the generation of large random surface gravity waves by a hinged wavemaker in a large-scale wave flume demonstrates excellent agreement between the desired theoretical spectral representation and the smoothed, measured spectral representation for two types of two-parameter theoretical spectra as a result of the lengthier realization made possible by the stacked FFT algorithm.
Abstract: A stacked inverse finite Fourier transform (FFT) algorithm is presented that will efficiently synthesize a discrete random time sequence of N values from only N/2 complex values having a desired known spectral representation. This stacked inverse FFT algorithm is compatible with the synthesis of discrete random time sequences that are used with the more desirable periodic-random type of dynamic testing systems used to compute complex-valued transfer functions by the frequency-sweep method. An application to the generation of large random surface gravity waves by a hinged wavemaker in a large-scale wave flume demonstrates excellent agreement between the desired theoretical spectral representation and the smoothed, measured spectral representation for two types of two-parameter theoretical spectra as a result of the lengthier realization made possible by the stacked FFT algorithm.

27 citations


Journal ArticleDOI
Lawrence R. Rabiner1
TL;DR: This paper shows how a similar approach can be used for sequences which are known to have only odd harmonics, and is shown to be essentially the dual of the known method for time symmetry.
Abstract: It is well known that if a finite duration, N-point sequence x(n) possesses certain symmetries, the computation of its discrete Fourier transform (DFT) can be obtained from an FFT of size N/2 or smaller. This is accomplished by first preprocessing the sequence, taking the FFT of the processed sequence, and then postprocessing the results to give the desired transform. In this paper we show how a similar approach can be used for sequences which are known to have only odd harmonics. The approach is shown to be essentially the dual of the known method for time symmetry. Computer programs are included for implementing the special procedures discussed in this paper.

26 citations


Journal ArticleDOI
TL;DR: A modified version of the Winograd-Fourier transform algorithm is presented for use in transforming real vectors, using real arithmetic and real storage of intermediate results throughout while retaining the economy of Winog rad's basic method.
Abstract: A modified version of the Winograd-Fourier transform algorithm is presented for use in transforming real vectors. The new algorithm uses real arithmetic and real storage of intermediate results throughout while retaining the economy of Winograd's basic method. The derivation of the transform is explained and some programming techniques are discussed and illustrated.

16 citations


Journal ArticleDOI
TL;DR: For certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform (DFT) is extended considerably by performing the cyclic convolution with the Mersenne prime number-theoretic transform developed originally by Rader.
Abstract: In this paper for certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform (DFT) is extended considerably. This is accomplisbed by performing the cyclic convolution, required by Winograd's method, with the Mersenne prime number-theoretic transform developed originally by Rader. This new algorithm requires fewer multiplications than either the standard fast Fourier transform (FFT) or Winograd's more conventional algorithm. However, more additions are required.

10 citations


Journal ArticleDOI
TL;DR: Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms, which produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.
Abstract: Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms. For periodic data adequately sampled, the fast Fourier transform (FFT) is used directly. With undersampled or aperiodic data, a Chebyshev interpolating polynomial is evaluated by means of the FFT to provide minimum deviation and distributed ripple. The merits of two kinds of Chebyshev series are compared. All the methods described produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.

Journal ArticleDOI
TL;DR: The program is an implementation of the optimal sorting algorithm of the author which allows a base-2 version of the Cooley-Tukey FFT algorithm efficient access to a mass store array.
Abstract: The program is an implementation of the optimal sorting algorithm of the author [8] which allows a base-2 version of the Cooley-Tukey FFT algorithm [2-4] efficient access to a mass store array. Optimal sorting for the mass storage FFT has been determined independently by DeLotto and Dotti [5, 6], but in the author's version the emphasis is on \"in-place\" array modification. This results in slightly higher mass store I /O than the minimum, but requires no additional mass store working space. The method is a logical extension of the work of Singleton [9] and Brenner [1]. The program computes in place the discrete Fourier transform of a onedimensional or a multidimensional array. In the one-dimensional case the transform is defined by

Journal ArticleDOI
TL;DR: The Fast Fourier Transform (FFT) for a step-like bounded function with unequal values at boundaries may be computed by using a convenient decomposition of the total curve into two elementary ones, one of them being a linear ramp as discussed by the authors.
Abstract: The calculation of the fast fourier transform (FFT) for a step-like bounded function with unequal values at boundaries may be performed by using a convenient decomposition of the total curve into two elementary ones, one of them being a linear ramp. The method may be generalized to functions having asymptotic tails which may be approximated by simple analytic functions, the theoretical FFT of which is known.

Proceedings ArticleDOI
Henri J. Nussbaumer1, P. Quandalle
01 Apr 1979
TL;DR: This paper presents two methods for computing discrete Fourier transforms (DFT) by polynomial transforms that are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).
Abstract: Polynomial transforms defined in rings of polynomials, have been introduced recently and shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper, we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).

Journal ArticleDOI
TL;DR: A multiddimensional fast-Fourier-transform algorithm is developed for the computation of multidimensional Fourier and Fourier-like discrete transforms; it has considerably less multiplications than the conventional fast-fourier -transform methods.
Abstract: A multidimensional fast-Fourier-transform algorithm is developed for the computation of multidimensional Fourier and Fourier-like discrete transforms; it has considerably less multiplications than the conventional fast-Fourier-transform methods.

15 Feb 1979
TL;DR: The new DFT algorithm of S. Winograd is developed and presented in detail and is applicable to any order which is a product of relatively prime factors from the following list: 2,3,4,5,7,8,9,16.
Abstract: The new DFT algorithm of S. Winograd is developed and presented in detail. This is an algorithm which uses about 1/5 of the number of multiplications used by the Cooley-Tukey algorithm and is applicable to any order which is a product of relatively prime factors from the following list: 2,3,4,5,7,8,9,16. The algorithm is presented in terms of a series of tableaus which are convenient, compact, graphical representations of the sequence of arithmetic operations in the corresponding parts of the algorithm. Using these in conjunction with included Tables makes it relatively easy to apply the algorithm and evaluate its performance.

01 Jan 1979
TL;DR: In this article, bounds on the minimum number of data transfers (i.e., loads and stores) required by WFTA (Winograd Fourier Transform Algorithm) and FFT programs are presented.
Abstract: Bounds on the minimum number of data transfers (i.e., loads and stores) required by WFTA (Winograd Fourier Transform Algorithm) and FFT programs are presented. The analysis is applicable to those general-purpose computers with a small number of general processor registers (e.g., the IBM370, PDP-11, etc.). It is shown that the 1008-point WFTA requires about 21% more data transfers than the 1024-point radix-4 FFT; on the other hand, the 120-point WFTA has about 22% fewer data transfers than the 128-point radix-2 FFT. Finally, comparisons of the 'total' program execution times (multiplications, additions, and data transfers, but not indexing or permutations) are presented.

Proceedings ArticleDOI
04 Sep 1979
TL;DR: A description of a machine built to use one of the new algorithms is given and the problems which were encountered are discussed and some possible structures of machines are suggested.
Abstract: When computing the Fourier Transform with a microprocessor, the speed and complexity of the algorithm which is used become especially important. The most frequently used algorithm has been the Fast Fourier Transform. More recently developed algorithms require fewer multiplications and about the same number of additions as the FFT. A comparison of these algor ithms is made and some possible structures of machines are suggested. A description of a machine built to use one of the new algorithms is given and the problems which were encountered are discussed.

Journal ArticleDOI
TL;DR: It is shown that there are position-dependent bounds on the error amplitude in the Fourier co-efficients, which means that the error statistics are position dependent and the earlier results on finite arithmetic effects in FFT calculation are inaccurate to that extent.
Abstract: For the decimation-in-frequency FFT algorithm using fixed point arithmetic, it is shown that there are position-dependent bounds on the error amplitude in the Fourier co-efficients. This means that the error statistics are position dependent and the earlier results on finite arithmetic effects in FFT calculation are inaccurate to that extent. These results lead to worst-case deterministic design of FFT processor.

Proceedings ArticleDOI
02 Apr 1979
TL;DR: The algorithm uses the fast Fourier transform to diagonalize and decouple the system of equations which results from the application of the least-squares criterion, and is perhaps an order of magnitude faster than the best iterative method.
Abstract: This paper describes a fast direct algorithm for obtaining least-squares phase estimates from arrays of noisy phase differences. The algorithm uses the fast Fourier transform (FFT) to diagonalize and decouple the system of equations which results from the application of the least-squares criterion. It is accurate and stable, and is perhaps an order of magnitude faster than the best iterative method. The effectiveness of the algorithm has been demonstrated by using it in connection with the Knox-Thompson speckle-imaging procedure to restore an optical object perturbed by simulated atmospheric turbulence.

Proceedings ArticleDOI
01 Apr 1979
TL;DR: This comparison shows that the relative time efficiency of the two algorithms in sequential computations generally carries over to cases where arithmetic parallelism is exploited.
Abstract: Arithmetic concurrencies, such as those found in special-purpose fast Fourier transform (FFT) hard-ware, are surveyed and categorized. Similar structures are then derived for the Winograd Fourier transform algorithm (WFTA). Relative time-efficiency plots are obtained for the 1024-point radix-4 FFT and the 1008-point WFTA as a function of the number of real arithmetic operations executable in parallel. This comparison shows that the relative time efficiency of the two algorithms in sequential computations generally carries over to cases where arithmetic parallelism is exploited.

Proceedings ArticleDOI
01 Apr 1979
TL;DR: Practical algorithms for computing the discrete Fourier transform (DFT) result from the Kronecker product of small N algorithms and it is shown that several options exist for indexing of the input and output of this DFT.
Abstract: Practical algorithms for computing the discrete Fourier transform (DFT) result from the Kronecker product of small N algorithms. This paper shows that several options exist for indexing of the input and output of this DFT. If the Chinese remainder theorem (CRT) is used to expand the output index, then an alternate integer representation (AIR) is shown to determine the input index. It is shown that the roles of the CRT and AIR can be reversed so that the CRT and AIR determine input and output indices, respectively. As a consequence of the indexing the DFT must be processed in subspaces whose dimensions are relatively prime.