Posibits, negabits, and their mixed use in efficient realization of arithmetic algorithms

Abstract

Positively weighted and negatively weighted bits (posibits, negabits) have been used in the interpretation of 2's-complement, negative-radix, and binary signed-digit number representation schemes as a way of facilitating the development of efficient arithmetic algorithms for various application domains. In this paper, we show that a more general view of posibits and negabits, along with their mixed use in any combination (using inverse encoding for negabits), unifies a number of diverse implementation schemes, while at the same time making the resultant designs more efficient by avoiding custom or modified hardware elements and restricting the implementation to the use of standard arithmetic cells. Such standard cells have been highly optimized and are continually improving due to their wide applicability. Other practical benefits of our formulation include facilitation of low-voltage and low-power design, again due to the widespread availability of standard cells in variants optimized for low-voltage operation or energy economy. Pedagogical benefits include more intuitive explanations for a number of widely used transformations, such as Booth's recoding and column compression.