Contributions of Graham Jullien and William Miller to Modified Quadratic Number System Arithmetic

Abstract

Quadratic Residue Number System (QRNS) arithmetic became very popular in the early 1980’s. QRNS allows one to compute the multiplication of two complex numbers with only two real multiplies as compared to the four multiplies required in normal complex multiplication. This is particularly useful in the computation of Fast Fourier Transforms (FFT), Complex Number Theoretic Transforms (CNTT) and digital filters with complex coefficients. However, QRNS requires the RNS moduli to be prime numbers of the form 4k + 1 or composite numbers with prime factors of that form. This restriction eliminates some of the most desirable moduli for RNS number systems. In 1986, Graham Jullien and William Miller introduced what they called Modified Quadratic Number Systems (MQRNS) which allowed any set of valid RNS moduli to be used in an MQRNS system in which two complex numbers could be multiplied with three real multiplications: one more that QRNS, but one less than normal complex multiplication. This MQRNS system lent itself well to applications in FFT, CNTT and digital filters with complex coefficients including complex heterodyne tunable filters.