Trinary Signed-digit Research Articles

Nonrecoded trinary signed-digit multiplication based on digit grouping and pixel assignment

A two-step trinary signed-digit (TSD) multiplication technique based on digit grouping and smart pixel assignment is proposed. In this technique, the partial products are generated directly without recoding the TSD numbers. This nonrecoding based TSD multiplication technique results in significant reduction in the number of symbolic substitution rules employed in the algorithm. Several spatial encoding schemes are introduced and their relative performance in the presence of additive noise is discussed.

Code conversion from signed-digit to complement representation based on look-ahead optical logic operations

We present, for the first time to our knowledge, a generalized lookahead logic algorithm for number conversion from signed-digit to complement representation. By properly encoding the signed-digits, all the operations are performed by binary logic, and unified logical expressions can be obtained for conversion from modified-signed-digit (MSD) to 2's complement, trinary signed-digit (TSD) to 3's complement, and quarternary signed-digit (QSD) to 4's complement. For optical implementation, a parallel logical array module using an electron-trapping device is employed and experimental results are shown. This optical module is suitable for implementing complex logic functions in the form of the sum of the product. The algorithm and architecture are compatible with a general-purpose optoelectronic computing system. (C) 2001 Society of Photo-Optical Instrumentation Engineers.

Trinary signed-digit arithmetic using an efficient encoding scheme

The trinary signed-digit (TSD) number system is of interest for ultrafast optoelectronic computing systems since it permits parallel carry-free addition and borrow-free subtraction of two arbitrary length numbers in constant time. In this paper, a simple coding scheme is proposed to encode the decimal number directly into the TSD form. The coding scheme enables one to perform parallel one-step TSD arithmetic operation. The proposed coding scheme uses only a 5-combination coding table instead of the 625-combination table reported recently for recoded TSD arithmetic technique.

Parallel optoelectronic trinary signed-digit division

The trinary signed-digit (TSD) number system has been found to be very useful for parallel addition and subtraction of any arbitrary length operands in constant time. Using the TSD addition and multiplication modules as the basic building blocks, we develop an efficient algorithm for performing parallel TSD division in constant time. The proposed division technique uses one TSD subtraction and two TSD multiplication steps. An optoelectronic correlator based architecture is suggested for implementation of the proposed TSD division algorithm, which fully exploits the parallelism and high processing speed of optics. An efficient spatial encoding scheme is used to ensure better utilization of space bandwidth product of the spatial light modulators used in the optoelectronic implementation.

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.

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

High-radix number systems enable higher information storage density, less complexity, fewer system components, and fewer cascaded gates and operations. A simple one-step fully parallel high-radix signed-digit arithmetic is proposed for parallel optical computing based on new joint spatial encodings. This reduces hardware requirements and improves throughput by reducing the space-bandwidth product needed. The high-radix signed-digit arithmetic operations are based on classifying the neighboring input digit pairs into various groups to reduce the computation rules. A new joint spatial encoding technique is developed to present both the operands and the computation rules. This technique increases the spatial bandwidth product (SBWP) of the spatial light modulators (SLMs) of the system. An optical implementation of the proposed high-radix signed-digit arithmetic operations is also presented. It is shown that our one-step trinary signed-digit (TSD) and quarternary signed-digit (QSD) arithmetic units are much simpler and better than all previously reported high-radix signed-digit techniques.

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.

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.