By Topic

Structural properties of complex residue rings applied to number theoretic Fourier transforms

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

1 Author(s)
M. Vanwormhoudt ; University of Ghent, Ghent, Belgium

The complex residue ring C(m) modulo m is first defined. Because of the existence of a ring isomorphism known as the Chinese remainder theorem (CRT), the study of C(m) can be limited to the cases where m = pe, p being a prime. C(pe) contains a multiplicative group, the group Q(pe) of the invertible elements. Q(pe) is shown to be the product of a group of order p2e-2and of a group R(pe), which is of order (p - 1)2when 4 divides (p - 1), and of order (p2- 1) when 4 is no divisor of (p - 1). It is shown that there exists an isomorphic mapping of Q(p) \longleftrightarrow R(p^{e}) . Consequently, the study of the orders of the elements of R(pe) can be reduced to studying those of Q(p). When 4 is not a divisor of (p - 1), Q(p) and R(pe) are cyclic groups of order (p2- 1). When 4 divides (p - 1), the elements of C(p) can be isomorphically mapped on Z(p) × Z(p), Z(p) being the set of real residue classes, mod p. In this case, the order of the elements of Q(p) and of R(pe) are limited to the divisors of (p - 1). When 4 does not divide (p - 1), all elements of R(pe) satisfy a set of orthogonality relations. This property also holds true for some of the elements of R(pe) when 4 divides (p - 1). The foregoing results are applied to number theoretic Fourier transforms in C(m). A necessary and sufficient condition is derived for N to be a possible transform length. It is shown that all reductions mod p\min{i}\max {e_{i}} of the transform factor where p\min{i}\max {e_{i}} represent the prime power factors of m, must belong to R(p\min{i}\max {e_{i}} and be of order N. Where Fermat number transforms (FNT) do not lead to transform lengths that are larger in C(m) than in Z(m), Mersenne number transforms result in a very large increase of the allowable values of N. The paper ends with a discussion on how a search procedure in Q(p) or in Z(p) allows to determine all available transform factors in Q(pe) for a given N.

Published in:

IEEE Transactions on Acoustics, Speech, and Signal Processing  (Volume:26 ,  Issue: 1 )