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.
Highlights
Addition is the most fundamental arithmetic operation in most processors
The n-bit modified signed-digit (MSD) addition of A and B can be carried out in three steps according to TW transformations in Table 1, and its computing processes are listed in Table 2 [29]
Theorem 4: All transformations Y, F, Y, F and S given in Table 7-13 have the characteristics of parallel carry-free three-step MSD addition
Summary
Addition is the most fundamental arithmetic operation in most processors. But the problem of carrying propagation from the least-significant-bit to the most-significant-bit in conventional binary addition inevitably affects the computing speed. The n-bit MSD addition of A and B can be carried out in three steps according to TW transformations, and its computing processes are listed in Table 2 [29]. Theorem 3: For any two MSD numbers A = anan−1 · · · a2a1 and B = bnbn−1 · · · b2b1, the cases yi−1 = fi = 1 and 1 ̄ cannot appear by applying any set transformations Y , F, Y , F of Table 9-13 to a and b successively. Theorem 4: All transformations Y , F, Y , F and S given in Table 7-13 have the characteristics of parallel carry-free three-step MSD addition. The work in this paper provides a theoretical basis for the design of the carry-free adder of the ternary optical computer
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