Fast Fourier Transforms: for fun and profit

@inproceedings{Gentleman1966FastFT,
  title={Fast Fourier Transforms: for fun and profit},
  author={W. Morven Gentleman and Gordon Sande},
  booktitle={AFIPS '66 (Fall)},
  year={1966}
}
The "Fast Fourier Transform" has now been widely known for about a year. During that time it has had a major effect on several areas of computing, the most striking example being techniques of numerical convolution, which have been completely revolutionized. What exactly is the "Fast Fourier Transform"? 

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References

SHOWING 1-10 OF 10 REFERENCES
An algorithm for the machine calculation of complex Fourier series
TLDR
Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
High-speed convolution and correlation
TLDR
A procedure for synthesizing and analyzing Fourier series for discrete periodic complex functions and computation times are proportional to log 2sub 2 as expressed in Eq.
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis
TLDR
This work has developed a direct method of solution involving Fourier analysis which can solve Poisson''s equation in a square region covered by a 48 x 48 mesh in 0.9 seconds on the IBM 7090.
Some Practical Considerations in the Realization of Linear Digital FiltersOn an Alternative Method of Calculating Covariance Functions
  • Proceedings of the Third Allerton Conference on Circuit and System Theory
  • 1965
Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs
  • New Jersey,
  • 1963
On an Alternative Method of Calculating Covariance Functions," unpublished
  • Princeton University,
  • 1965
Stockham, "High Speed Convolution and Correlation," AFIPS, volume
  • Spring Joint Computer Conference, Spartan Books, Washington,
  • 1966
Hamming, Numerical Analysis for Scientists and Engineers, McGraw-Hill
  • New York,
  • 1962