New Symbolic Substitution Algorithms for Optical Arithmetic using Signed -Digit Representation

In this paper, we present a new approach to designing optical digital arithmetic systems. We present new arithmetic algorithms based on a modified signed -digit number representation. New Signed -digit symbolic substitution rules are introduced to implement them. These signed -digit arithmetic algorithms are well suited for optical implementation because, of the confined carry propagation within adjacent digits. We present an optical architecture of such an arithmetic processor. The proposed optical arithmetic processor can potentially achieve 0(102) to O(103) improvement in speed as compared with conventional electronic arithmetic processors. i_.INTRODUCTON The inherent parallelism, high speed, noninterfering communication, and wide bandwidth of optics show hope for significantly improving the speed of arithmetic computations in future optical computers. In this paper, we proceed in a two-step approach: first, new algorithms are developed for digital arithmetic operations that are amenable to optical implementation; then, we present a new arithmetic architecture and discuss its implementation and performance based on the state -of -the -art optical technology. Several number representations have been investigated in the past for optical arithmetic. Residue number system has been considered'. The advantage of usig residue numbers lies in performing carry -free addition, subtraction, and multiplication. However, this system suffers from the difficulties of performing division, an comparing two numbers. The residue number system is an integer field, however the results of division are not always integers and therefore do not always have a residue representation. Furthermore, it is not easy to determine the sign after a subtraction operation. A second number system known as DMAC has been introduced2. Its major drawback is the need for analog -to-digital conversion required to convert the mixed binary to pure binary numbers. This postprocessing step offsets the speed gained by optical processing. hi this paper, we use a modified signed -digit (SD) number representation originally proposed by Avizienis3, and lately introduced to optics by Drake et al4 to perform optical arithmetic computations. The SD number representation is a redundant system with a digit set of {1, 0,1 }, where 1 stands for -1. The redundancy overcomes the strong interdigit dependency that results in carry propagation manifested in conventional binary representation. Carry propagation is then limited to two adjacent digits in an SD system. This makes it possible to perform addition and subtraction of two SD numbers of any word length in parallel. Holographic optical elements, prisms, and optical bistable devices were proposed for implementing the SD addition and subtraction optically4. Mirsalehi and Gaylord3 proposed a direct table look -up method for implementing the SD addition. In this paper, we implement the SD addition, subtraction, multiplication, and division using optical symbolic substitution technique extended from the work of Huangs. We introduce a new set of bit -wise, signeddigit, symbolic substitution rules. The use of signed -digit number representation in conjunction with these new symbolic substitution rules for implementing optical operations will yield an optical arithmetic processor with a tremendous speed improvement over existing electronic counterparts. 2. SIGNED -DIGIT ARITHMETIC ALGORITHMS The modified SD number system has a digit set of {1, 0,1 }. Given an SD number Y = yn -iyn -2 ' yoy -1 y -m, the algebraic value of Y is evaluated as : i=n-1 Y= x2i, yiE{I,0,1} (1) In this number system, there is no need for an explicit sign digit, in fact yn_1 determines the sign of Y. Hereafter, we concentrate on integer SD numbers for addition /subtraction and multiplication and on normalized fractions for division. We present below algorithms for various SD operations. The SD addition is generalized from Avizienis3. The SD multiplication and SD division algorithms are newly developed. 90 / SPIE Vol. 880 High Speed Computing (1988) lic tit r t ti usi g Sig e igit