Symmetrically recoded modified signed-digit optical addition and subtraction

Abstract

Symmetrically recoded modified signed-digit number algorithms are proposed for carry-free arithmetic. The superiority of the new algorithms is established over all other two-step-based symbolic-substitution optical implementations.

Efficient two-step digit-set-restricted modified signed-digit algorithm based on optoelectronic shared content-addressable memory

“…Therefore, only 12 minterms ͑10 minterms in the first step and 2 minterms in the second one͒ are required. In comparison, Cherri's SRMSD schemes require 12 minterms for the positive outputs and 12 complemented minterms for the negative outputs, 6 while our algorithm requires only 12 minterms for all outputs. Additionally, the SRMSD schemes require six four-variable minterms in the second step.…”

“…However, a conventional binary system limits computation speed because of its strong interdigit dependency due to carry propagation from the least-significant-bit position to the most-significant-bit position. Consequently, various parallel algorithms, especially those using signed-digit ͑SD͒ number representations, [1][2][3][4][5][6][7][8][9][10][11][12] have been developed in the past two decades to overcome the problem. The redundant nature of the SD representations enables addition of two arbitrary-length operands to be completed in less than three steps.…”

“…Various carry-free MSD addition algorithms have been reported, and they are usually classified in terms of the involved step number, i.e., three-step, two-step, and onestep algorithms. [1][2][3][4][5][6][7][8][9] Many optical architectures for the implementations of those algorithms have also been reported, including symbolic-substitution-based architectures. [2][3][4][5][6][7][8] In all symbolic substitution schemes, the most essential task is to minimize the number of the required minterms ͑or symbolic substitution rules͒, and thus system complexity.…”

“…Another important two-step approach using 16 minterms is based on the recoding of MSD numbers. 5 Recently, Cherri 6 proposed two symmetrically recoded MSD ͑SRMSD͒ schemes, which need only 12 minterms. In most of the previously reported MSD algorithms, minterms for output digit 1 are complement to those for 1.…”

Principle and design of ternary optical accumulator implementing M-k-B addition

“…Cherri proposed symmetrically recoded modified signeddigit two-step optical addition and subtraction. 15 Huang et al 16 proposed a one-step MSD parallel addition and subtraction, in which the whole of three adjacent digits is divided into 10 groups and each group is judged. Qian et al proposed a two-step MSD addition and subtraction algorithm based on binary logic arithmetic using electron-trapping device, and presented one-step digit-set-restricted modified signed-digit adder.…”