New Algorithm for Signed Integer Comparison in $\{2^{n+k},2^{n}-1,2^{n}+1,2^{n\pm 1}-1\}$ and Its Efficient Hardware Implementation

Abstract

Sign detection and magnitude comparison are two difficult operations in Residue Number System (RNS). Existing algorithms mainly tackle either problem independently. Magnitude comparison in RNS is typically solved by parity check or transcoding the residues into weighted representation. Comparison of signed integers in residue domain is supplemented by separately designed RNS sign detector. In this paper, a radically different quantization approach for comparing signed integers in the four-moduli supersets, $S_{1}\equiv\{2^{n+k},2^{n}-1,2^{n}+1,2^{n+1}-1\}$ and $S_{2}\equiv\{2^{n+k},2^{n}-1,2^{n}+1,2^{n-1}-1\}$ , is proposed. The dynamic range of the target moduli set is quantized into equal divisions, and the ranks of the divisions resided by the integers in comparison are identified and compared from their residue representations. The sign of a residue representation can be directly extracted from the most significant bit of its rank, or with simple logic function if it resides in the middle rank. Comparing with the best existing signed magnitude comparator applicable to these two moduli sets, our synthesis results on 65 nm CMOS technology show that the proposed design is at least 22.48% smaller, 19.08% faster and 27.66% more energy-efficient for $S_{1}$ and 18.75% smaller, 16.41% faster and 23.30% more energy-efficient for $S_{2}$ .