Abstract

Modified signed-digit binary numbers have demonstrated the promise of carry-free addition. In this method, the to-be-added pair of numbers are mapped into an intermediate pair of numbers in such a way that when the latter numbers are added, they do not result in carry generation. The corresponding recognition and substitution operations, referred to as symbolic substitution, can be performed in parallel at all digit positions; thus, the method fits elegantly with optical computing architecture. Three-step, two- step, and single step¹,² carry-free addition based on optical symbolic substitution schemes have already been worked out for the binary system. In this paper, we extend the carry-free addition and borrow-free subtraction scheme to accommodate signed-digit trinary numbers by deriving the necessary substitution rules to perform addition/sub- traction in two steps. A number of possible sets of rules exist for addition and subtraction. The reduced minterms are derived for a possible two-or three-step holographic content addressable memory implementation.¹,²

© 1990 Optical Society of America