Balanced $(3+2\log n)\Delta G$ Adders for Moduli Set $\{{2}^{n+1},2^{n}+2^{n-1}-1,2^{n+1}-1\}$

Abstract

Residue number systems (RNS) are characterized by fast modular arithmetic and low power dissipation. Numerous RNS applications take advantage of moduli set ${\tau }=\{{2}^{{{n}}}{-1,}{2}^{{{n}}},\,\,2^{{{n}}}+1\}$ , with nearly ${2}^{3n}$ dynamic range and fast parallel-prefix adders. However, the ${2}^{{{n}}}+1$ channel is ${4}{\Delta }{{G}}$ slower ( ${\Delta }{{G}} =$ simple 2-input gate delay). To remedy such speed imbalance and accommodate higher dynamic ranges, other moduli forms have joined $\{{2}^{{{n}}},{2}^{{{n}}}-1\}$ , with only slightly slower adders (e.g., ${2}^{{{n}+1}}{-1} (n={2}^{{\text {h}}}){,}{2}^{{{n}}}{-}{2}^{q}{-1({q} , and ${2}^{{{n}}}{-3}$ , with at most ${2\Delta {G}}$ more parallel-prefix delay). However, the more the number of moduli, the harder and more costly becomes the required reverse convertor, while otherwise increasing the channel widths ${n}$ may incur some speed loss. Nevertheless, in the present work, the 3-moduli set ${\tau }^{+}=\{{2}^{{{n}+1}}{,}{2}^{{{n}}}{+}{2}^{n-1}{-1,} {2}^{{{n}+1}}{-1}\}$ is presented, with almost $6\times $ dynamic range than that of the aforementioned $\tau $ , where the corresponding parallel-prefix adders are as fast as those for the ${2}^{{{n}}}$ and ${2}^{{{n}}}{-1}$ channels. The required nontrivial forward convertor for modulo ${2}^{{{n}}}+{2}^{{{n}-1}}{-1}$ and the reverse convertor for the new 3-moduli set are also designed. Moreover, a new 3-input parallel-prefix node is designed and incorporated, as appropriate for ${n}= 2^{\text {h}}$ , within the employed parallel-prefix networks with the overall advantage of $2\Delta {G}$ speed-up. Improvements of proposed designs are confirmed via circuit synthesis.