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A NEW PROPOSAL OF BINARY DATA ADDITION AND SUBTRACTION SCHEME WITH MULTIVALUED NUMBER SYSTEM Partha Pratim Das and Dibyendu Giri Department of Engineering Science, Haldia Institute of Technology Haldia, Purba Medinipur, West Bengal-721 657, India E-mail: dppratim@yahoo.co.in Abstract: In the decade of eighty one of the most important fields of researches in optical computing was very much exposed. It is optical parallel data processing, with ‘carry’ and ‘borrow’–less technique. The main difficulty in parallel arithmetic operations of binary or decimal numbers lies with the accommodation of carry and borrows. To evaluate this problem “Mixed Signed Digit System”, “Modified Mixed Signed Digit” etc. has been proposed earlier. In this chapter another number system named “Multivalued (Tristate) Number System” system is proposed for parallel arithmetic operations with binary number system. Conceptual implementation of addition by optical technique has been proposed also. 1. INTRODUCTION also be adopted in the arithmetic operations suggested here [5]. To eliminate the difficulties stated above “Mixed Signed Digit Number (MSD) System”, “Modified 2. MULTIVALUED (TRISTATE) NUMBER Mixed Signed Digit (MMSD) Number System”, SYSTEM “Trinary / Ternary/ Tristate Number System” etc. have Multivalued (tristate) Number System was been proposed earlier from the various corners of the proposed first in 1988 for handling parallel and carry- parallel computation research area [1-2]. In this chapter free arithmetic operations. Since the data or numbers completely a new technique of carry and borrow-free are usually available in binary form, therefore the arithmetic operation (data addition) is proposed for process of symbolic substitutions and the substitution parallel computation. In such connection a conceptual of data by signed digit numbers are of a great implementation of digital data addition with importance to access the parallel processing “Multivalued (tristate) Number System” by an optical mechanism satisfactorily. technique also has been proposed here. Any binary number having the digits (bits) One of the advantages of using optics for {0,1} and radix 2 can be converted to a tristate number computing lies in its massively parallel processing capability [3]. To utilize this advantage one must with digits {1,0, 1 }, and radix 2, where 1 is “Most evolve a suitable method for carry and borrow-free Significant Bit”(MBS). The technique of the arithmetic operation [4]. It has been shown that parallel conversion is as follows: carry and borrow-free addition and subtraction can be Any N bits binary number can be expressed as accomplished either optically or electronically or N electro-optically by representing the respective numbers / data in Modified Signed Digit (MSD) ∑C n2 n −1 (1) number representation. There are many other processes n =1 where symbolic substitutions of data are also suggested Where Cn takes the bit values of binary number 0 or 1. for parallel handling of carry and borrow-free MTN is a radix 2 number system where “Most operations. Significant Bit” (MSB) is only 1 and rest bit is either 0 Optical shadow casting technique has much or 1 . Therefore 1+1 possibility in data addition does usefulness in digital optical computing for the above not arise, when bit-wise addition is performed between purpose. Any binary operation can be handled by addend and augend. proper coding in shadow casting technique. In this chapter the coding processes of data in shadow casting Now I will describe the technique of technique are exploited for addition of two images in conversion of a tristate number from its binary parallel. It has also been shown that spatial translations equivalent. A tristate number is expressed as of image in both x and y-axis are possible in shadow M casting system. This spatial translation mechanism may 2N + ∑C m2 m −1 (2) 1 where Cm takes the value either 0 or 1 . 1 bit is nothing Table-1: Addition of two numbers X and Y but –1. The conversion of any binary number to its Augend (Y) 13 1101 respective multivalued (tristate) number can be done in 100 1 1 (Decimal) (Binary) (Tristate) parallel. For example a binary number 11011011, decimal equivalent of 219, can be represented by Addend (X) 12 1100 01100 “Multivalued (tristate) Number System” as (Decimal) (Binary) (Binary) Result of 25 11001 111 1 1 100 1 00 1 0 1 . Addition (Decimal) (Binary) (Tristate) The multivalued (tristate) number has been proposed for both decimal and binary number system. 4. OPTICAL IMPLEMENTATION OF BIT-WISE If a decimal or binary number is expressed in DATA ADDITION AND SUBTRACTION IN multivalued (tristate) number system, the carry or MULTIVALUED (TRISTATE) NUMBER borrow-free arithmetic operations are possible and this SYSTEM is very much important for parallel processing to construct a optical computing architecture. Fundamental approach for computing with optical signal lies in the representation of binary data 3. DATA ADDITION AND SUBTRACTION format, where one (1) is represented by the presence of SCHEME WITH MTN SYSTEM light and zero (0) is considered as the absence of that. Now data addition scheme with multivalued The data are generally recorded into a 2D format with (tristate) number system is discussed with an example the help of spatial light modulator/ by optical shutters (Table-1). For this process one have to convert either or by non-linear optical elements and the operations are the augend or addend into multivalued (tristate) carried out in 2D, so that parallelism is established. In number and then process of parallel addition is carried this section we will show the improved experimental out. It is evident that no carry and borrow is involved implementation of multivalued (tristate) number based in this process of addition as well as subtraction. arithmetic addition process. Suppose we have two data in binary form as First we discuss on an optical spatial encoding X=1100, Y=1101. Now we convert any one of the technique of two binary data and then how the binary data either augend or addend into multivalued superimposed coded data is used suitably to obtain arithmetic subtraction will be discussed. In fig. 1 we (tristate) number with the digits of {1,0, 1 }. Let we have shown spatial coded representation of the bits 0 & convert Y=1101 into multivalued (tristate) number. In 1 in two strings of data X and Y, which are to be multivalued (tristate) number system, Y=1101 becomes operated. Here each cell is divided into two sub-cells. 100 1 1 . It should be mentioned that in multivalued The hatched sub-cells indicate no light and white (tristate) number system the bits of the number are 1,0 portions indicate presence of light. In such way two and 1 , but the radix is 2, whereas the most significant binary number of X and Y are coded. bit (MSB) is strictly 1. In this multivalued (tristate) number based addition process, the possibilities of bit- 1 0 wise additions are 0(X)+0(Y)=0, 0(X)+1(Y)=1, 1(X)+ 1 (Y)=0 and 0(X)+ 1 (Y)= 1 , when we consider that the Y data is represented in multivalued (tristate) X number. It is important to note that the MSB of the result of addition is MSB of Y data. Y From the example, we have the final result of addition in multivalued (tristate) number system. In binary addition scheme the question of carry and borrow must Fig. 1. Optical binary data in coded form for two come but in multivalued (tristate) number system this different inputs X and Y. process is carry and borrow-free and hence it is suitable Now if a coded bit of X and that of Y are for optical parallel computation. Following the same super-imposed, then the following possible procedure subtraction can be done easily. combinations are seen, which are given in fig.2 (a). There are four possible combinations for two binary numbers X & Y. There are X=0, Y=0; X=0, Y=1; X=1, Y=0 & X=1, Y=1. If we take only the respective contributions of light from the sub-cell number 1, and 4 only in the superimposed structure then we can get the resultant bits of subtraction, which is shown, in fig. Therefore first we should encode spatially the 2(b). Here all the four combinations of bit wise bits 0 & 1 in data X, which is a ordinary binary number subtraction in binary / tristate are 1-1=0, 0-1= 1 , 0-0=0 (addend) and then encode the bits 0, 1 & 1 spatially for and 1-0=1. The result of subtraction is in tristate. data Y, which is in multivalued (tristate) number (augend). In fig. 3 we have shown schematic coded X=1, Y=0 X=0, Y=1 diagram of the digits 0 &1 in X and 0, 1 & 1 in Y. Here each cell is divided into eight (8) equal sub-cells. The hatched sub-cells indicate no light while the white 1 3 1 3 sub-cells indicate presence of light. In such way digits of X & Y [Y in multivalued (tristate) number] are encoded. 2 4 2 4 0 1 X=1, Y=1 X=0, Y=0 1 3 1 3 1 0 1 2 4 2 4 Fig.2(a). Four different possible spatially coded Fig. 3. Representation of digits for inputs X & Y in structures of inputs, X and Y. coded forms of multivalued (tristate) number addition. Now if we superimposed a coded digit of X & that of Y, then the following four possible combinations come, which are shown in fig. 4(a). If we 1 1 take only the respective contributions of light from sub-cell number 5, 7 & 4 in the superimposed Tristate Outputs configuration then we can get the resultant bits of addition represented spatially by the output digits 0,1 & 0 0 1 in the form of spatially coded pattern as shown in fig. 4(b). It is clear that in the spatially coded output bit Fig. 2(b). Squares 1 and 4 in each combined structure of the result of addition we have three sub-cells taken gives the optically coded di-bit answer for subtraction. from the above superimposed position. When all the Now we discuss about the all-optical sub-cells receive light then it indicates the implementation of addition scheme in multivalued representation of 0. Similarly (tristate) number system following the same process of spatial encoding of data. We have seen earlier that for X=0, Y=0 X=1, Y=0 addition any one of the two binary numbers must be 1 1 2 3 4 2 3 4 converted into multivalued (tristate) number, while the other may remain in binary. To implement this addition 5 6 7 8 5 6 7 8 scheme in all-optical approach we require the successful obtaining of the fundamental needs 0+0=0, X=1, Y=1 X=0, Y=1 0+1=1, 1+ 1 =0 & 0+ 1 = 1 , because only four 1 2 3 4 1 2 3 4 possibilities come in multivalued (tristate) number system addition. As multivalued (tristate) number 5 6 7 8 5 6 7 8 system is a radix 2 based number system where the MSB is strictly one (1) and rest other bits are either 0 Fig.4(a). Four different superimposed structures of X & or 1 as necessary, therefore 1+1 possibility in data Y for multivalued(tristate) number addition. addition does not come at all, when bit-wise addition is carried between addend and augend. 5. CONCLUSION 0 1 A newer scheme of parallel (carry & borrow- free) arithmetic addition with multivalued (tristate) number system is described in this chapter. The scheme discussed in this chapter is an all-optical approach, which runs in parallel. Only by encoding the inputs spatially and by simple superimposition of the input 0 data we can achieve the scheme, where the output bits 1 of addition are received also by spatial coding. There are two important points in connection to its implementation. First of which is that for superimposition of the images of the two spatially coded input bits, we can exploit a lens based system or Tristate Outputs fibre optic mapping system. Second is that, the result of the data addition is obtained in plain tristate number Fig.4(b). The MSB 1 of the result of multivalued system, not in multivalued (tristate) number system. (tristate) number based addition (the result is in The parallel conversion of tristate number to its tristate). equivalent binary form can be done. However, if the conversion of binary number to its equivalent when light is present only in the first two right most multivalued (tristate) number and the conversion from sub-cells, it indicates 1 and when the light exists only a tristate number (result) to this equivalent binary form in the left most sub-cells then it indicates 1 . are done parallely then the whole process of addition Y=1 in M.S.B. (where the inputs & outputs both are in binary) will come to be a parallel running system. 1 2 3 4 The scheme described above is a process of 5 6 7 8 bit wise addition between augend and addend. This part is fully parallel with the proper exploitation of the inherent parallelism of optical signal. The main limitation is diffraction. 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