Complex approximations using algebraic integers
Richard A. Games · IEEE Transactions on Information Theory · 1985
The problem of approximating complex numbers by elements ofZ[\omega], the algebraic integers ofQ(\omega), where\omegais a primitiventh root of unity, is considered. The motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete Fourier transform. Smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument. Scale factors involved are reduced from\alphato\sqrt{\alpha}for this case with roughly the same percentage errors. The case of sixteenth roots of unity gives even better range reductions and is considered only briefly.