Introduction to Cooley and Tukey (1965) An Algorithm for the Machine Calculation of Complex Fourier Series

Fourier (frequency) analysis is important in statistics, mathematics, and all the hard sciences (as in x-ray crystallography), and in signal processing, so Fast Fourier Transforms (FFTs) are widely applicable. For example, Costain (1974) said the FFTs are regarded as an economic breakthrough in geo­physics, especially in seismology.

[1]  O. L. Davies,et al.  Design and analysis of industrial experiments , 1954 .

[2]  Charles M. Rader,et al.  Number theory in digital signal processing , 1979 .

[3]  J. Kane,et al.  Kronecker Matrices, Computer Implementation, and Generalized Spectra , 1970, JACM.

[4]  I. Good.  Saddle-point Methods for the Multinomial Distribution , 1957 .

[5]  I. J. Good,et al.  A new finite series for Legendre polynomials , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  L. P. Williams,et al.  Joseph Fourier: The Man and the Physicist , 1975 .

[7]  I. J. Good,et al.  The Relationship Between Two Fast Fourier Transforms , 1971, IEEE Transactions on Computers.

[8]  I. Good.  Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables , 1963 .

[9]  M. Kline.  Mathematical Thought from Ancient to Modern Times , 1972 .

[10]  I. J. Good,et al.  The Interaction Algorithm and Practical Fourier Analysis: An Addendum , 1960 .

[11]  H. Cramér.  Random variables and probability distributions , 1937 .

[12]  V. Ralph Algazi,et al.  A Unified Treatment of Discrete Fast Unitary Transforms , 1977, SIAM J. Comput..

[13]  On a method of trend elimination. , 1949, Biometrika.

[14]  G. Tolstov.  Fourier Series , 1962 .

[15]  I. Good.  On the Serial Test for Random Sequences , 1957 .

[16]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[17]  I. Good,et al.  On the inversion of circulant matrices. , 1950, Biometrika.

[18]  L. J. Savage,et al.  Probability and the weighing of evidence , 1951 .

[19]  J. S. Hunter.  The Inverse Yates Algorithm , 1966 .

[20]  Don H. Johnson,et al.  Gauss and the history of the fast Fourier transform , 1985 .

[21]  I. Good,et al.  The inversion of the discrete gauss transform , 1979 .

[22]  W. E. Thomson.  Ernie—A Mathematical and Statistical Analysis , 1959 .

[23]  I. J. Good,et al.  Analogues of Poisson's Summation Formula , 1962 .

[24]  C. Lanczos,et al.  Some improvements in practical Fourier analysis and their application to x-ray scattering from liquids , 1942 .

[25]  M. Schroeder.  Number Theory in Science and Communication , 1984 .

[26]  I. Good.  C285. A survey of the use of the fast fourier transform for computing distributions , 1987 .

[27]  I. Good,et al.  A Table of Discrete Fourier Transform Pairs , 1977 .

[28]  I. Good,et al.  The poisson and chi-squared approximations as compared with the true upper-tail probability of Pearson's X2 for equiprobable multinomials , 1986 .

[29]  Klaus Hinkelmann,et al.  Design and Analysis of Experiment , 1975 .

[30]  I. J. Good,et al.  The fast calculation of the exact distribution of pearson's chi-squared and of the number of repeats within the cells of a mltltinomial by using a fast fourier transform , 1981 .

[31]  I. Good.  The serial test for sampling numbers and other tests for randomness , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[32]  F. Yates.  Design and Analysis of Factorial Experiments , 1958 .

[33]  I. J. GOOD,et al.  Polynomial Algebra: an Application of the Fast Fourier Transform , 1969, Nature.

[34]  Peter D. Welch,et al.  Historical notes on the fast Fourier transform , 1967 .

[35]  S. Burr.  The Unreasonable Effectiveness of Number Theory , 1993 .

[36]  Kenneth L. Caspari,et al.  Computer techniques in image processing , 1972 .

[37]  S. Winograd.  On computing the Discrete Fourier Transform. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[38]  I. Good.  Random motion on a finite Abelian group , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[39]  C. Runge,et al.  Vorlesungen über Numerisches Rechnen , 1924 .

[40]  R. Bracewell.  The Fourier Transform and Its Applications , 1966 .

[41]  R. Singleton.  An algorithm for computing the mixed radix fast Fourier transform , 1969 .

[42]  Morton Nadler,et al.  Pattern recognition engineering , 1993 .

[43]  C. Loan.  Computational Frameworks for the Fast Fourier Transform , 1992 .

[44]  I. J. Good,et al.  A Paradox Concerning Rate of Information , 1958, Inf. Control..

[45]  P. Laplace.  Théorie analytique des probabilités , 1995 .

[46]  James W. Cooley,et al.  The 1968 Arden house workshop on fast Fourier transform processing , 1969 .

[47]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[48]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[49]  R. Garwin,et al.  Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon , 1957 .

[50]  I. Good.  C138. The fast calculation of the exact distribution of pearson's chi-squared and of the number of repeats within the cells of a multinomial by using a fast fourier transform: an addendum , 1982 .

[51]  Harry C. Andrews,et al.  A Generalized Technique for Spectral Analysis , 1970, IEEE Transactions on Computers.