Implementation of complex number theoretic transforms using quadratic residue number systems

Abstract

Very recently, the Quadratic Residue Number System (QRNS) has been introduced [6], [7]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. The conjugate ring has the remarkable property that both addition and multiplication are performed component-wise. Complex multiplication only requires two base field multiplications and zero additions; this contrasts to the requirement of four multiplications and two additions, or three multiplications and five additions for the Gaussian integer ring. The operations are performed over sub-rings, isomorphic to the conjugate ring, and the results mapped to the conjugate ring via the Chinese Remainder Theorem isomorphism. The primary restriction is the limited form of the moduli set for RNS computations. The QRNS has since been generalized for any type of moduli set with an increase in multiplications from 2 to 3 and the resulting number system has been termed the Modified Quadratic Residue Number System (MQRNS) [3], [4]. In [4] the direct FIR filter architecture and bit-slice architecture for FIR and recursive digital filters have been presented using the QRNS and MQRNS. In this paper, the computation of the Complex Number Theoretic Transform (CNTT) and the hardware implementation of a radix-2 butterfly structure, using high-density ROM arrays, are presented. This paper shows that both the QRNS and MQRNS require almost the same amount of hardware for the implementation of the butterfly structure. The computation of Cyclic Convolution in both the QRNS and MQRNS is discussed.