Arithmetic for Optical Computing

The interest in parallel binary and non-binary computer arithmetic in digital computing was initiated with the pioneering works of Avizienis. 1 The parallel nature of such modified binary systems prompted researchers to adopt the modified signed-digit~MSD! algorithm for optical computing. 2 Around the same time, non-binary systems such as multiple valued logic~MVL ! ~Ref. 3! also achieved prominence both in optics and digital computing. Since then a large number of papers have been published in optical computer arithmetic. This special section is an attempt to capture current research in computer arithmetic for optical computing. The five major areas that are presented in this section are MSD-based algorithm and systems, optimization of MVL, novel architectures for binary optical computing, high accuracy analog optical system implementations, and system studies for fault-tolerance and accuracy. Some of the papers may have overlap of two or more areas with one primary focus; they are pointed out in the following discussion. The largest cluster of papers appears in the area of signed-digit arithmetic and its implementation. A number of different techniques for addition, multiplication and division are proposed by several authors. The number systems addressed include redundant binary, MSD binary, negabinary, MSD trinary, recoded trinary and MSD quaternary. In terms of number of steps, addition/subtraction in single, dual and triple step has been proposed. While the MSD number system leads to higher information density, if the number of steps is reduced, the truth tables may become humongous, which may impose challenging requirements on the actual implementation. Techniques for reducing the cost of such implementations have been addressed by some authors. Proposed implementations include space-variant logic array, correlator ~composite and pseudo-inverse filter ! and non-holographic content addressable memory~CAM! using electron-trapping material. Several authors have proposed novel algorithms and their possible optical implementations while others have suggested implementations and/or optimization on known algorithms. The first paper in the area of signed-digit arithmetic by Li et al. presents negabinary arithmetic operations for addition, subtraction and multiplication and implements them using electron trapping material. A carry free addition technique in signed-digit negabinary ~SDN! is presented with a conversion technique from SDN to normal negabinary. Zhang and Karim propose modified two-step, one-step, canonical and three-input algorithms for addition of redundant binary numbers and provide architecture and encoding for corresponding optical space-variant implementation. Cherri demonstrates single step trinary and quaternary signed-digit circuits. In general, the reduction in step increases the complexity of the truth table. However, Cherri overcomes the problem by smart digit grouping to reduce the number of rules and an intelligent pixel encoding to implement the system within a certain space-bandwidth product. Huang, Itoh and Yatagai propose a new technique for high-speed 2-D data array addition and multiplication based on binary MSD addition and digit-decomposition-plane representation. Huang, Itoh and Yatagai generate all the partial products in parallel and propose to add them using an MSD adder tree. It is interesting to note that they perform multiplication operation using five elementary operations such as bitwise product, duplication, shifting, masking and magnification. In the next paper, Alam introduces trinary division technique based on recoded trinary addition and multiplication. The proposed implementation uses a pseudo-inverse filter correlator. The last two papers in this group by Ahmed, Awwal and Power and by Zhang and Karim propose novel implementations of trinary and binary MSD algorithm. Ahmed, Awwal and Power implement an MSD trinary adder using composite phase-only-filter correlator architecture. In this framework, the truth table rules are Special Section Guest Editorial