A NEW PROPOSAL OF BINARY DATA ADDITION AND SUBTRACTION by nfj14094

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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 .
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
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
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
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. To avoid this limitation we
may use optical fibre as wave-guide with the proper
control of the input and output cell sizes. We should
take care about the format of both the input structures
1                                  of X & Y. If the dimension of the structures is not same
then problem in super-impositions will arise.
Fig.5. The output form of representation of MSB of Y
coded data.                                                                     REFERENCES
Thus we can exploit all the four possibilities,    [1] R.P.Bocker, B.L.Drake, M.E.Lasher and
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(tristate) number based carry & borrow-free addition.        subtraction using optical symbolic substitution”, Appl.
As 1+1 combination will not at all comes so we have          Opt., vol.25, pp.2456-2457, 1986.
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the multivalued (tristate) number can be represented.        2001.
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case of the MSB representation is shown in fig. 5. We        by using a ternary digit representation technique in
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substitution”, Opt.Eng., vol.28,pp.364, 1989.

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