DocumentCode
923022
Title
The use of finite fields to compute convolutions
Author
Reed, Irving S. ; Truong, T.K.
Volume
21
Issue
2
fYear
1975
fDate
3/1/1975 12:00:00 AM
Firstpage
208
Lastpage
213
Abstract
A transform is defined in the Galois field of 
elements 
, a finite field analogous to the field of complex numbers, when 
is a prime such that (--1) is not a quadratic residue. It is shown that the action of this transform over is equivalent to the discrete Fourier transform of a sequence of complex integers of finite dynamic range. If is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.
elements , a finite field analogous to the field of complex numbers, when is a prime such that (--1) is not a quadratic residue. It is shown that the action of this transform over is equivalent to the discrete Fourier transform of a sequence of complex integers of finite dynamic range. If is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.Keywords
Convolution; Galois fields; Number-theoretic transforms; Arithmetic; Discrete Fourier transforms; Discrete transforms; Dynamic range; Fast Fourier transforms; Filtering; Fourier transforms; Galois fields; Helium; Roundoff errors;
fLanguage
English
Journal_Title
Information Theory, IEEE Transactions on
Publisher
ieee
ISSN
0018-9448
Type
jour
DOI
10.1109/TIT.1975.1055352
Filename
1055352
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