Stored-transfer representations with weighted digit-set encodings for ultrahigh-speed arithmetic

Abstract

Redundant representations play an important role in high-speed computer arithmetic. One key reason is that such representations support carry-free addition, that is, addition in a small, constant time, independent of operand widths. The implications of stored-transfer representation of digit sets and the associated addition schemes, as an extension of the stored-carry concept to redundant number systems, on the speed and cost of arithmetic algorithms, are explored. Two's-complement digits as the main part and any two-valued digit (twit) in place of a stored carry are allowed, leading to further broadening of the generalised signed-digit representations. The characteristics of the digit sets, possibly not having zero as a member, that allow for most efficient carry-free addition, are investigated. Circuit speed is gained from storing or saving, instead of combining through addition, the interdigit transfers generated during the carry-free addition process. Encoding efficiency is gained from using a twit-transfer set encoded by one logical bit, where more bits would otherwise be needed to represent a transfer value.