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 ABSTRACT
A modified version of the Fast Fourier Transform is developed and described. This version is well adapted for use in a special-purpose computer designed for the purpose. It is shown that only three operators are needed. One operator replaces successive pairs of data points by their sums and differences. The second operator performs a fixed permutation which is an ideal shuffle of the data. The third operator permits the multiplication of a selected subset of the data by a common complex multiplier.
If, as seems reasonable, the slowest operation is the complex multiplications required, then, for reasonably sized date sets—e.g. 512 complex numbers—parallelization by the method developed should allow an increase of speed over the serial use of the Fast Fourier Transform by about two orders of magnitude.
It is suggested that a machine to realize the speed improvement indicated is quite feasible.
The analysis is based on the use of the Kronecker product of matrices. It is suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.
REFERENCES
Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.
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COOLEY, J. W., LEWIS, P. A. W., AND WELCH, P. D. Historical notes on the Fast Fourier Transform. IEEE Trans. A U-15 (June 1967), 76-79.
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.... AND TUKEY, J .W . An algorithm for the machine calculation of colnplex Fourier series. Math. Comput. 19, 90 (April 1965), 297-301.
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STrocKnaM, T. G., JR. High-speed convolution and correlation. Proe. AF{PS 1966 Spring Joint Comput. Conf., Vol. 28, pp. 229-233.
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BINGHAM, C., GODFREY, M. D., AND TUKEY, J.W. Modern techniques of Power spectruin estimation. (Unpublished.)
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GENTLEMAN, W. M., AND SANDE, G. Fast Fourier Transforms- for fun and profit. Proe. AFIPS 1966 Fall Joint Comput. Conf., Vol. 29, pp. 563-578.
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SINGLETON, R. C; A method for computing the Fast Forelet Transform with auxiliary memory and limited high-speed storage. IEEE Trans. A U-15 (June 1967), 91-97.
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COOLEY, J. W., LEWIS, P. A. W., AND WELCH, P.D. Application of the Fast Fourier Transform to computation of Fourier integrals, Fourier series, and convolution integrals. IEEE' Trans. A U-15 (June 1967), 79-84.
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HELMS, H. D. Fast Fourier Transform method of computing difference equAtions and simulating filters. IEEE .Trans. A U-i5 (June 1967), 85-90.
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G-AE SUHUICMMUTEE ON MEASUREMENT CONCEPTS. What is the Fast Fourier Transforms IEEE Trans. A U-I5 (June 1967), 45-55.
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PEASE, M.C. Melhods of Malrix Algebra. Academic Press, New York, 1965, Ch. XIV.
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CITED BY 40
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Peter A. Milder , Mohammad Ahmad , James C. Hoe , Markus Püschel, Fast and accurate resource estimation of automatically generated custom DFT IP cores, Proceedings of the internation symposium on Field programmable gate arrays, February 22-24, 2006, Monterey, California, USA
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Parimala Thulasiraman , Kevin B. Theobald , Ashfaq A. Khokhar , Guang R. Gao, Multithreaded algorithms for the fast Fourier transform, Proceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures, p.176-185, July 09-13, 2000, Bar Harbor, Maine, United States
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