Modified signed-digit trinary arithmetic by using optical symbolic substitution

Designs of optoelectronic trinary signed-digit multiplication by use of joint spatial encodings and optical correlation

Trinary signed-digit (TSD) symbolic-substitution-based (SS-based) optical adders, which were recently proposed, are used as the basic modules for designing highly parallel optical multiplications by use of cascaded optical correlators. The proposed multiplications perform carry-free generation of the multiplication partial products of two words in constant time. Also, three different multiplication designs are presented, and new joint spatial encodings for the TSD numbers are introduced. The proposed joint spatial encodings allow one to reduce the SS computation rules involved in optical multiplication. In addition, the proposed joint spatial encodings increase the space-bandwidth product of the spatial light modulators of the optical system. This increase is achieved by reduction of the numbers of pixels in the joint spatial encodings for the input TSD operands as well as reduction of the number of pixels used in the proposed matched spatial filters for the optical multipliers.

Recoded and nonrecoded trinary signed-digit adders and multipliers with redundant-bit representations

Highly-efficient two-step recoded and one-step nonrecoded trinary signed-digit (TSD) carry-free adders-subtracters are presented on the basis of redundant-bit representation for the operands' digits. It has been shown that only 24 (30) minterms are needed to implement the two-step recoded (the one-step nonrecoded) TSD addition for any operand length. Optical implementation of the proposed arithmetic can be carried out by use of correlation- or matrix-multiplication-based schemes, saving 50% of the system memory. Furthermore, we present four different multiplication designs based on our proposed recoded and nonrecoded TSD adders. Our multiplication designs require a small number of reduced minterms to generate the multiplication partial products. Finally, a recently proposed pipelined iterative-tree algorithm can be used in the TSD adders-multipliers; consequently, efficient use of all available adders can be made.

Optoelectronic Recoded and Nonrecoded Trinary Signed-Digit Adder that uses Optical Correlation

A symbolic-substitution-based optical numeric processor that uses recoded and nonrecoded trinary signed-digit (TSD) number representations is proposed. Also, we propose new joint spatial encodings for the TSD numbers that reduce the symbolic-substitution computation rules involved in the processor. Optoelectronic implementation of the proposed recoded adder is feasible. Also, the nonrecoded TSD addition can be performed optically in two steps. Both the proposed recoded and nonrecoded adders are more compact than a recently reported modified signed-digit counterpart and use fewer correlators and spatial light modulators.

Parallel optical negabinary arithmetic based on logic operations

On the basis of signed-digit negabinary representation, parallel two-step addition and one-step subtraction can be performed for arbitrary-length negabinary operands. The arithmetic is realized by signed logic operations and optically implemented by spatial encoding and decoding techniques. The proposed algorithm and optical system are simple, reliable, and practicable, and they have the property of parallel processing of two-dimensional data. This leads to an efficient design for the optical arithmetic and logic unit.

Parallel optical computing using recoded trinary signed-digit numbers

We propose a simple recoded trinary signed-digit number representation for parallel optical computing. This technique performs multidigit carry-free addition and borrow-free subtraction in constant time, using only 50% of the, minterms required in the most recently reported trinary signed-digit arithmetic technique. One may use a single-step optoelectronic or a two step all-optical architecture to implement the proposed technique.

Efficient trinary signed-digit symbolic arithmetic

A new technique for high-speed trinary signed-digit (TSD) arithmetic that uses optical symbolic substitution is presented. This technique performs multibit carry-free addition and borrow-free subtraction in constant time. The proposed TSD scheme utilizes the minimum number of minterms compared with the previously reported modified signed-digit and TSD number systems.