Gauss and the History of the Fast Fourier Transform

IEEE ASSP Magazine

Volumn 1, Issue 4, 1984, Pages 14-21

  Heideman, Michael T ^a   Johnson, Don H ^a   Burrus, C Sidney ^a  

^a RICE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL MODELS;

MATHEMATICAL TRANSFORMATIONS;

EID: 0021510651     PISSN: 07407467     EISSN: None     Source Type: Journal    
DOI: 10.1109/MASSP.1984.1162257     Document Type: Article

References (47)

1. Cooley, J. W. and Tukey, J.W., “An algorithm for the machine calculation of complex Fourier series,” Math. Comput., vol. 19, no. 2, pp. 297–301, April 1965. Reprinted in Digital Signal Processing, L. R. Rabiner and C.M. Rader, Eds., pp. 223–227, New York: IEEE Press, 1972.
2. Brigham, E.O., The Fast Fourier Transform, Englewood, NJ: Prentice-Hall, Inc., 1974.
3. Good, I. J., “The interaction algorithm and practical Fourier analysis,” J. R. Statist Soc. B, vol. 20, no. 2, pp. 361–372, 1958. Addendum in J. R. Statist. Soc. B, vol. 22, no. 2, pp. 373–375, 1960.
4. Rudnick, P., “Note on the calculation of Fourier series,” Math. Comput, vol. 20, no. 3, pp. 429–430, July 1966.
5. Danielson, G. C. and Lanczos, C., “Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids,” J. Franklin Inst., vol. 233, nos. 4 and 5, pp. 365–380 and 435–452, April and May 1942.
6. Cooley, J.W., Lewis, P. A.W., and Welch, P.D., “Historical notes on the fast Fourier transform,” IEEE Trans. Audio Electroacoust., AU-15, no. 2, pp. 76–79, June 1967. Reprinted in Proc. IEEE, vol. 55, no. 10, pp. 1675–1677, Oct. 1967. Also reprinted in Digital Signal Processing, L. R. Rabiner and C. M. Rader, Eds., pp. 260–262, New York: IEEE Press, 1972.
7. Runge, C., “Uber die zerlegung empirisch gegebener periodischer funktionen in sinuswellen,” Z, Math. Phys., vol. 48, pp. 443–456, 1903.
8. Runge, C., “Uber die zerlegung einer empirischen funktion in sinuswellen,” Z, Math. Phys., vol. 52, pp. 117–123, 1905.
9. Goldstine, H.H., A History of Numerical Analysis from the 16th Through the 19th Century, pp. 249–253, New York, Heidelberg, and Berlin: Springer-Verlag, 1977.
10. Gauss, C.F., “Nachlass: Theoria interpolationis methodo nova tractata,” pp. 265–303, in Carl Friedrich Gauss, Werke, Band 3, Göttingen: Königlichen Gesellschaft der Wissenschaften, 1866.
11. Burkhardt, H., “Trigonometrische interpolation,” in ch. 9, Encyklopädie der Mathematischen Wissen-schaften, vol. 2, part 1, 1st half, pp. 643–693, Leipzig: B. G. Teubner, 1899–1916.
12. Heideman, M.T. and Burrus, C.S., “A Bibliography of Fast Transform and Convolution Algorithms II,” Technical Report Number 8402, Electrical Engineering Dept., Rice University, Houston, TX 77251-1892, Feb. 1984.
13. Thomas, L. H., “Usinga computer to solve problems in physics,” in Applications of Digital Computers, Boston: Ginn, 1963.
14. Winograd, S., “On computing the discrete Fourier transform,” Proc. Nat Acad. Sci. USA, vol. 73, no. 4, pp. 1005–1006, April 1976.
15. Runge, C. and könig, H., “Vorlesungen uber numerisches rechnen,” Die Crundlehren der Mathematischen Wissenschaften, vol. 11, pp. 211–237, Berlin: Julius Springer, 1924.
16. Stumpff, K., Tafeln und Aufgaben zur Harmonischen Analyse und Periodogrammrechnung, Berlin: Julius Springer, 1939.
17. Whittaker, E.T. and Robinson, G., The Calculus of Observations; pp. 260–284, London: Blackie and Sons, Limited, 1924.
18. Thompson, S. P., “A new method of approximate harmonic analysis by selected ordinates,” Proc. Phys. Soc. London, vol. 23, pp. 334–343, 1911.
19. Thompson, S.P., “Note on a rapid method of harmonic analysis,” Proc. Phys. Soc. London, vol. 19, pp. 443–453, 1904.
20. Darwin, G. H. and Adams, J. C., “The harmonic analysis of tidal observations,” Brit. Assoc. Report, pp. 49–118, 1883. Reprinted in G. H. Darwin: Scientific Papers Volume I: Oceanic Tides and Lunar Disturbances of Gravity, pp. 1–69, Cambridge: University Press, 1907.
21. Smith, A., Admiralty Scientific Manual on “Deviations of the Compass”, 5th ed., London: Hydrographic Office, Admiralty, 1874.
22. Strachey, R., “On the computation of the harmonic components of a series representing a phenomenon recurring in daily and yearly periods,” Proc. Roy. Soc. London, vol. 42, no. 251, pp. 61–79, 1887.
23. Everett, J. D., “On a method of reducing observations of underground temperature, with its application to the monthly mean temperatures of underground thermometers at the Royal Edinburgh Observatory,” Trans. Roy. Soc. Edin., vol. 22, no. 2, pp. 429–439, 1860.
24. Thomson, W., “On the reduction of observations of underground temperature; with application to Professor Forbes’ Edinburgh observations, and the continued Carlton Hill Series,” Trans. Roy. Soc. Edin., vol. 22, no. 2, pp. 405–427, 1860. Reprinted in Mathematical and Physical Papers, Vol. 3 of William Thomson, Lord Kelvin, pp. 261–290, London: C. J. Clay and Sons, Cambridge Univ. Press, 1890.
25. Smith, A. and Sabine, E., “Contributions to terrestrial magnetism —no. VIII, memorandum on elements of reduction,” Phil. Trans. Roy. Soc. London, vol. 136, no. 3, pp. 347–355, 1846.
26. Smith, A., Instructions for the Computation of a Table of the Deviations of a Ship’s Compass, From Deviations Observed on 4, 8, 16, and 32 Points, and for the Adjustment of the Table on a Change of Magnetic Latitude, London: Gt. Brit. Hydrographic Office, 1850.
27. Smith, A., Supplement to the Practical Rules for Ascertaining the Deviations of the Ship’s Compass Caused by the Ship’s Iron, Being Instructions for the Computation of a Table of the Deviation of the Ship’s Compass from Observations Made on 4, 8, 16, or 32 Points (Second Edition) and a Graphic Method of Correcting the Deviations of a Ship’s Compass, London: J.D. Potter, 1855.
28. Carlini, F., Sulla Legge Delle Variazioni Orarie del Bar-ometro, Modena: Presso la Tipographia Camerale, 1828. Also in Societa Italiana della Scienze, Rome, Memorie di Matematica e di Fisica, vol. 20, p. 198, 1828.
29. Hansen, P. A., Mémoire Sur la Détermination des Perturbations Absolues dans des Ellipses d’une ExcentricM et d’une Inclinaison Quelconques. Paris: Bachelier, 1835.
30. Markel, J.D,, “FFT pruning,” IEEE Trans. Audio Electroacoust., AU-19, no. 4, pp. 305–311, Dec. 1971.
31. Euler, L., Introductio in Analysin Infinitorum, Lausanne: Marc-Michel Bousquet et Comp., 1748. Reprinted in Leonhardi Euleri Opera Omnia, Series)., vol. 8, A. Krazer and F. Rudio, Eds., Leipzig and Berlin: B.G. Teubner, 1922.
32. Euler, L., “Observationes generales circa series quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur,” Nova Acta Academiae Scientiarum Petropolitanae, vol. 7, pp. 87–98, 1789. Reprinted in Leonhardi Euleri Opera Omnia, Series I, vol. 16, Section 1, Commentationes Analyticae ad Theoriam Serierum Infinitarum Pertinentes, vol. 3, Section 1, C. Boehm, Ed., pp. 163–177, Leipzig and Berlin: B.G. Teubner, 1933.
33. Euler, L., “Methodus facilis inveniendi series per sinus cosinusve angulorum multiplorum procedentes quarum usus in universa theoria astronomiae est amplissimus,” Nova Acta Academiae Scientiarum Petropolitanae, vol. 11, pp. 94–113, 1793. Reprinted in Leonhardi Euleri Opera Omnia, Series I, Vol. 16, Section 1, Commentationes Analyticae ad Theoriam Serierum Infinitarum Pertinentes, vol. 3, section 1, C. Boehm, Ed., pp. 311–332, Leipzig and Berlin: B.G. Teubner, 1933.
34. Clairaut, A.-C., “Mémoire sur I’orbite apparente du soleil autour de la terre, en ayant égard aux perturbations produites par les actions de la lune et des planétes principals,” Memoires de Mathematique et de Physique de l’Académie Royäle des Sciences, no. 9, pp. 521–564, 1754.
35. Lagrange, J.L., “Solution de différents problèmes de calcul integral,” Miscellanea Taurinensia ou Mélanges de Turin, vol. Ill, 1762–1765. Reprinted in Oeuvres de Lagrange, vol. 1, J. A. Serret, Ed., pp. 469–668, Paris: Gauthier-Villars, 1867.
36. Bernoulli, D., “Réflexions et édaircissemens sur les nouvelles vibrations des cordes,” Mémoires de I’Académie Royale des Sciences et Belles Lettres, Berlin, 1753.
37. Burkhardt, H., “Trigonometrische reihen und integrale (bis etwa 1850),” in ch. 12, Encyklopädie der Mathematischen Wissenschaften, vol. 2, part 1, 2nd half, pp. 819–1354, Leipzig: B.G. Teubner, 1904–1916.
38. Lagrange, J. L., “Recherches sur la nature et la propagation du son,” Miscellanea Taurinensia ou Melanges de Turin, vol. I, no. I-X, pp. 1–112, 1759. Reprinted in Oeuvres de Lagrange, vol. 1, J.A. Serret, Ed., pp. 39–148, Paris: Gauthier-Villars, 1867.
39. Dunnington, G.W., Carl Friedrich Gauss: Titan of Science, pp. 398–404, New York: Exposition Press, 1955.
40. Bracewell, R. N., The Fourier Transform and Its Applications, p. 205, New York: McGraw-Hill, 1978.
41. Burrus, C. S., “Index mappings for multidimensional formulation of the DFT and convolution,” IEEE Trans. Acoust., Speech, Sig. Process., ASSP-25, no. 3, pp. 239–242, June 1977.
42. Merzbach, U.C., Carl Friedrich Gauss: A Bibliography, Wilmington, DE: Scholarly Resources Inc., 1984.
43. Fourier, J.B.J., Théorie Analytique de la Chaleur, Paris: F. Didot,1822. English translation in The Analytical Theory of Heat, translated by Alexander Freeman, Cambridge: University Press, 1878.
44. Herivel, J., Joseph Fourier: The Man and the Physicist, p. 149, Oxford: Clarendon Press, 1975.
45. Hope, L. L., “A fast Gaussian method for Fourier transform evaluation,” Proc. IEEE, vol. 63, no. 9, pp. 1353–1354, Sept. 1975.
46. Good, I. J., “Inversion of the discrete Gauss transform,” Applic. Anal., vol. 9, no. 3, pp. 205–218, Oct. 1979.
47. Huang, T. S., “How the fast Fourier transform got its name,” Computer, vol. 4, no. 3, p. 15, May–June 1971.