Complex digital signal processing using quadratic residue number systems

Recently, the quadratic residue number system (QRNS) has been introduced [6], [7] which allows the multiplication of complex integers with two real multiplications. The restriction is that the number system has either all prime moduli of the form 4K + 1, or composite numbers with prime factors of that form. If an increase in real multiplications from two to three can be tolerated, then the restriction can be lifted to allow moduli of any form: the resulting number system is termed the modified quadratic residue number system (MQRNS). In this paper, the MQRNS is defined, and residue-to-binary conversion techniques in both the QRNS and MQRNS are presented. Hard-ware implementations of multiplication intensive, complex nonrecursive, and recursive digital filters are also presented in this paper where the QRNS and MQRNS structures are realized using a bit-slice architecture.