A Generalized Reverse Conversion Algorithm for Six-Moduli Set

Abstract

Residue Number System has emerged as an alternative number system with advantages in many real-life systems including in digit signal processing devices. Computational systems built on residue number system require both forward and reverse conversion processes. These converters respectively convert a given integer into its corresponding residues and calculate the original integer from its residues. While forward conversion is pretty straight forward, reverse conversion poses challenges often requiring difficult procedures. Much of residue number system research has therefore been devoted to design and implementation of efficient reverse conversion algorithm. The Chinese Reminder Theorem and the Mixed-Radix Conversion are the two popular ones. The Chinese Reminder Theorem results in complex circuitry that requires difficult computation involving large modulo-M values. The Mixed-Radix Conversion offers simplicity in designs although its steps are sequential. This paper proposes a generalized reverse conversion algorithm tailored for a six-moduli set with a large dynamic range. This innovative algorithm minimises the difficult multiplicative inverse operations found in the traditional reverse conversion methods paving the way for a more efficient reverse conversion processes for systems that requires high dynamic ranges. The new algorithm has been meticulously evaluated numerically on a proposed six-moduli set $\left\{2^{n+1}-1,2^{2 n}+1,2^{2 n+1}-1,2^{3 n}+1,2^{3 n+1}-1,2^{4 n}+1\right\}$ for even values of , to ensure its correctness and simplicity. The approach holds great promise for enhancing the development of reverse converters allowing the expansion of the landscape of residue number system.