Four subroutines compute one-dimensional and multi-dimensional Fourier transforms for real data, multi-dimensional complex Fourier transforms, and multi-dimensional sine, cosine and sine-cosine transforms. Subroutines use Cooley-Tukey fast Fourier transform. In all but one-dimensional case, transforms are calculated in up to six dimensions

Krogh, F. T.

English

NASA Technical Reports Server

May 1971	 Brief 71-10133 
NASA TECH BRIEF 
NASA Pasadena Office
	
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be directed to the Technology. Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Multi-Dimensional Real Fourier Transform 
The problem: 
To compute the one-dimensional and multi-dimen-
sional Fourier transforms for real data, the multi-
dimensional complex Fourier transforms, and the 
multi-dimensional sine, cosine, and sine-cosine trans-
forms. 
The solution: 
Four subroutines, each of which performs the 
required calculation for one of the four problems 
given. Each subroutine uses the Cooley-Tukey Fast 
FOurier Transform (FFT), and, in all but the one-
dimensional case, the transforms are calculated in up 
to six dimensions. 
How it's done: 
The Cooley-Tukey algorithm is an efficient com-
putational procedure for evaluating expressions of 
the form:
N-I	 j0,1 ... N-1 
Xj =
	
aW, w =e21N 
k=0	 i2=-1 
where x and a k are complex, and N is composite. 
If the calculations were to be carried out in a 
straightforward manner, with an operation defined as 
one complex multiplication and addition, a total of 
N(N- 1)operations would be required; i.e., N- lopera-
tions for each value of J . The Cooley-Tukey al-
gorithm, however, organizes the calculations in such 
a way that only (N 1092N) operations are required, 
which represents a significant savings for large N. 
Notes: 
1. These programs are written in FORTRAN V for 
use on the UN! VAC-1108 computer. 
2. The specific subroutines are as follows: NPO-
11648, Multi-Dimensional Real Fourier Trans-
form; NPO-I 1649, One-Dimensional Real Fourier 
Transform; NPO- 11651, Multi-Dimensional Com-
plex Fourier Transform; and NPO-11652, Multi-
Dimensional Sine, Cosine, and Sine-Cosine 
Transforms. 
3. Inquiries concerning these programs should specify 
the program desired, and may be directed to: 
COSMIC 
Barrow Hall 
University of Georgia 
Athens, Georgia 30601 
Reference: B71-10133 
Patent status: 
No patent action is contemplated by NASA. 
Source: Fred T. Krogh of

Jet Propulsion Laboratory

under contract to

NASA Pasadena Office

(NPO-11648, 11649, 11651, 11652) 
Category 09 
As 
NW
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19710000132 2020-03-17T02:24:24+00:00Z


Multi-dimensional real Fourier transform

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