Complex digital signal processing over finite rings

Very recently, the quadratic residue number system (QRNS) has been introduced [8], [9]. It is, in fact, a rediscovery of earlier work [29]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. This conjugate ring has the remarkable property that both addition and multiplication are performed componentwise. The operations are performed over subrings, isomorphic to the conjugate ring, and the results are mapped to the conjugate ring via the Chinese remainder theorem (CRT). The QRNS has since been generalized for any type of moduli set, with an inherent dynamic range reduction, and has been termed the quadratic-like residue number system (QLRNS) [7]. An alternate form removes the dynamic range reduction but requires an increase in multiplications from two to three; this system has been termed the modified quadratic residue number system (MQRNS) [3], [4]. This tutorial paper is a companion to another paper in this special issue [32]; it consolidates work on quadratic system implementations, with special emphasis on the modified system. The paper discusses, in some detail, the quadratic implementation of the two forms of complex convolution, which are naturally defined over finite rings, or fields, and which incur no scaling overhead. A new notation is introduced that eliminates confusion over the several mappings that are required in the quadratic representation, and initial work on the implementation of finite ring computational elements, suitable for VLSI fabrication, is also presented.