Modified signed-digit trinary arithmetic by using optical symbolic substitution

Abstract

Carry-free addition and borrow-free subtraction of modified signed-digit trinary numbers with optical symbolic substitution are presented. The proposed two-step and three-step algorithms can be easily implemented by using phase-only holograms, optical content-addressable memories, a multichannel correlator, or a polarization-encoded optical shadow-casting system.

Classified one-step high-radix signed-digit arithmetic units

“…Thus, our proposed TSD and QSD adder/subtracter units are much simpler and use fewer computation rules than any previously reported techniques. 10,13,17,18,19,24 Optical implementation of the proposed arithmetic can be carried out using the incoherent correlation scheme. Further, note that our proposed TSD and QSD adder/subtracter can also be implemented using matrix-multiplication-based techniques.…”

“…Using nonconventional number representations 2 ͑nonbinary͒ to design fast arithmetic units has gained much attention in recent years. Several nonbinary number representation schemes such as multiplevalued fixed radix-number, 3,4 residue number, 5-8 redundant number, 9 and signed-digit [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] were reported in the past decade to implement efficient arithmetic operations.…”

“…Among the various proposed signed-digit and higher order signed-digit number system arithmetics, three-step, 11,13 two-step, [14][15][16][17][18]24 and one-step 20,23 carry-free addition and borrow-free subtraction of signed-digit numbers are performed by checking a pair ͑pairs͒ of reference digits. Modified signed-digit ͑MSD͒, for radixϭ2, and high-radix-based signed-digit number systems such as the trinary signeddigit ͑TSD͒ and the quaternary signed-digit ͑QSD͒ number systems enable higher information storage density, less complexity, fewer system components, and fewer cascaded gates and operations.…”

Symmetrically recoded quaternary signed‐digit arithmetic using a shared content‐addressable memory

“…Higher radix-based signed-digit number systems, such as the quaternary signed-digit ͑QSD͒ number system, allow higher information storage density, less complexity, fewer system components, and fewer cascaded gates and operations. 8,9 Various optical architectures using n-step, 10 three-step, 11,12 two-step, 9,13 and one-step 3,14,15 carry-free addition have been reported. In general, computation speed and circuit complexity increase as the number of computation steps decreases.…”

“…A two-step scheme appears to be a prudent choice in terms of computation speed and storage complexity. 5,9 Recently, a two-step quaternary signed-digit addition technique has been reported 16 that performs carryfree addition and borrow-free subtraction by checking a pair of reference digits. This technique 16 increases the computation speed but requires a huge number of six-variable minterms for each output digit.…”