Characteristics of Parallel Carry-Free Three-Step MSD Additions

Abstract

Since the modified signed digital (MSD) redundant representation was proposed in the 1950s, lots of achievements have been made in MSD arithmetic. By inspecting the processes of the parallel carry-free three-step MSD addition, the transformations for such additions are studied in detail in this paper. The characteristics for parallel carry-free three-step MSD addition are proposed and the correctness is proved. Then seven groups of transformations that have characteristics of three-step MSD addition are presented. These groups of transformations are dual or self-dual, and some of them have simpler forms than the typical transformations consisting of $T, W, T', W', T_{2}$ . The general design mode of parallel carry-free three-step MSD additions and its applications in ternary optical computer (TOC) are further proposed. At the same time, single adder (multi-adder) reconstruction mode, processor bits allocation strategy, and light path diagrams are given. The optical experiments of MSD additions for three groups of MSD addition transformations show that the results of these transformations are correct. This work provides the theoretical basis for the design of the ternary optical computer adder.

Keywords

• Ternary optical computer
• modified signed digit (MSD) representation
• ternary logic transformation
• parallel computing
• carry-free addition