Optimality of the Fast Fourier transform
@article{Papadimitriou1979OptimalityOT, title={Optimality of the Fast Fourier transform}, author={Christos H. Papadimitriou}, journal={J. ACM}, year={1979}, volume={26}, pages={95-102} }
A graph-theoretic model for a class of linear algorithms computing the discrete Fourier transform of sequences of length a power of 2, the mformat~on flow network, is presented The information flow network correspondmg to the fast Fourier transform IS shown to be umquely optimal in tim class with respect to a naturally defined costÂ
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References
SHOWING 1-10 OF 13 REFERENCES
On computing the Discrete Fourier Transform.
- Computer ScienceProceedings of the National Academy of Sciences of the United States of America
- 1976
New algorithms for computing the Discrete Fourier Transform of n points are described, which use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.
The Linear Complexity of Computation
- Computer Science, MathematicsJACM
- 1975
An additive degree of freedom is defined, which turns out to be an exact measure of the complexity of computation of a family 7 of linear forms in r variables over a field.
Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform
- Computer ScienceJACM
- 1973
A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound <italic>c</italic> can be assigned to the modulus of the…
The Design and Analysis of Computer Algorithms
- Computer Science
- 1974
This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
An algorithm for the machine calculation of complex Fourier series
- Computer Science
- 1965
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.
Fast Convolution using fermat number transforms with applications to digital filtering
- Computer Science
- 1974
The structure of transforms having the convolution property is developed and an implementation on the IBM 370/155 is presented and compared with the fast Fourier transform (FFT) showing a substantial improvement in efficiency and accuracy.
AL What IS the fast Fourier transform~
- IEEE Trans Audio and Electroacoust A
Fast convolution using Fermat number transforms lEEE Trans Acoust
- Speech, Szgnal Processing, A SSP-22,
- 1974
The Design and Analysts of Computer Algorithms
- The Design and Analysts of Computer Algorithms
- 1974