Abstracts
A technique is described for the rapid Fourier transform of large series of numbers. The technique takes advantage of the fact that most digital series are highly factorizable by the number 2, which permits the use of the F.F.T. algorithm. Using two magnetic tape units, or alternatively magnetic disk facilities, very large series can be transformed efficiently with only modest computer facilities. For the transformation of odd-valued series the Thomas Prime-Factor and Gentleman and Sande algorithms are treated in detail.
Apresenta-se neste trabalho uma técnica de transformação rápida de Fourier aplicada a uma longa série de valores numéricos. A técnica tira partido do fato de que a grande maioria das séries digitalizadas é, em geral, suscetível de fatoração onde aparece frequentemente o fator 2, o que permite o emprego do algorítmo da transformação rápida de Fourier (F.F.T.). Com o emprego de duas fitas magnéticas ou discos, pode ser efetuada eficientemente a transformação de longas séries em computadores de modesta memória. O algorítmo de fatores primos de Thomas e o de Gentleman e Sande são, respectivamente, tratados em detalhe, na transformação de séries com numero ímpar de valores.
A hybrid algorithm for the rapid Fourier transform of extensive series of data
A. S. Franco; N. J. Rock
Instituto Oceanográfico da Universidade de São Paulo
SYNOPSIS
A technique is described for the rapid Fourier transform of large series of numbers. The technique takes advantage of the fact that most digital series are highly factorizable by the number 2, which permits the use of the F.F.T. algorithm.
Using two magnetic tape units, or alternatively magnetic disk facilities, very large series can be transformed efficiently with only modest computer facilities.
For the transformation of odd-valued series the Thomas Prime-Factor and Gentleman and Sande algorithms are treated in detail.
RESUMO
Apresenta-se neste trabalho uma técnica de transformação rápida de Fourier aplicada a uma longa série de valores numéricos. A técnica tira partido do fato de que a grande maioria das séries digitalizadas é, em geral, suscetível de fatoração onde aparece frequentemente o fator 2, o que permite o emprego do algorítmo da transformação rápida de Fourier (F.F.T.).
Com o emprego de duas fitas magnéticas ou discos, pode ser efetuada eficientemente a transformação de longas séries em computadores de modesta memória.
O algorítmo de fatores primos de Thomas e o de Gentleman e Sande são, respectivamente, tratados em detalhe, na transformação de séries com numero ímpar de valores.
Full text available only in PDF format.
Texto completo disponível apenas em PDF.
ACKNOWLEDGEMENTS
The authors gratefully wish to acknowledge the assistance of the Instituto de Física da Universidade de são Paulo in permitting extensive us e of the IBM 360/44 computer.
The research was carried out whilst the junior author was a visiting professor to the Instituto Oceanográfico with the assistance of the "Conselho Nacional de Pesquisas" and the "Fundação de Amparo à Pesquisa do Estado de São Paulo".
(Received 7/6/1971)
APPENDIX I
It is necessary to derive some general expression involving the operator "Mod" in order to simplify expression (4f).
From the definition itself of A Mod N, it follows that
Now, if α and β are the remainders of the division of integers A and B, respectively, by N, we can write:
thus
AB = INJN + αJN + βIN + αβ
but, if K is the quotient of the integer division of αβ by N then
αβ = KN + γ 
γ = (αβ) Mod N
and
AB (INJ + αJ + βI + K) N + γ
thus
or, according to (b)
From (b)
A + B = (I + J) N + (α + β)
but, if M and α are the quotient and the remainder, respectively of the division of α + β by N, it follows that
α + β = MN + δ δ = (α + β) Mod N
thus
A + B = (I + J + M) N + δ 
δ = (A + B) Mod N
consequently
or, according to (b)
Finally, if
Expressions (a), (d), (f) and (g) are all we need to effect all the developments.
APPENDIX II
FLOW DIAGRAMS and COMPUTER PROGRAMS
Click to enlarge
Click to enlarge
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- WATT, J. M. 1959. A note on the evaluation of trigronometric series. Computo J., 1.4, p. 162 Apud CARTWRIGHT, D. E. & CATTON, D. B. 1963. On the Fourier analysis of tidal observations. Int. hydrogr. Rev., vol. XL, no. 1, p. 113.125.
Publication Dates
-
Publication in this collection
12 June 2012 -
Date of issue
1971
History
-
Received
07 June 1971
