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The International Journal of EngineeringAnd Science (IJES)||Volume|| 1 ||Issue|| 1 ||Pages|| 18-21 ||2012||ISSN: 2319 – 18...
Multiplication another Hue And Crythe pertial product of the process is less tahn the                 E.  Booth’s Multi pl...
Multiplication another Hue And Cry                   I.    MOTIVATION                                          In the prop...
Multiplication another Hue And Cry             III.      Future Scope     In the proposed process of multip licat ion weno...
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The International Journal of Engineering and Science


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The International Journal of Engineering and Science

  1. 1. The International Journal of EngineeringAnd Science (IJES)||Volume|| 1 ||Issue|| 1 ||Pages|| 18-21 ||2012||ISSN: 2319 – 1813 ISBN: 2319 – 1805 Multiplication another Hue And Cry Barun Biswas 1, Samar Sen Sarma2, Krishnendu Basuli3 1 CVPR Un it, Indian Statistical Institute, 203 B.T Road Kolkata-700108, India 2 Department of Co mputer Science and Engineering, University of Calcutta, 92, A.P.C. Road, Kolkata –700 009, India. 3 Dept. of Co mputer Science, West Bengal State University, India-----------------------------------------------------------------Abstract--------------------------------------------------------Multi plication process is not only bounded only in mathematics. It is also applied in many other fields wherea sequence of summing or additi on is to be applied. Multi plication is nothing but a sequence of addi tionperformed repeatedl y. Our basic approach is to consider as many bits of multi plier as possible so that thetotal number of partial product. In this paper we tried to i mplement such a techni que where we can considermore than one two bi ts(compared to Booth’s process of multi plication) of the multi plier to decrease the totalnumber of parti al product with the other operation like addi tion, shift and complement.Key Words: Partial product, Shift operation, Comp lement operation, Booth’s process of multiplication, Shift andadd, repeated addition, Div ide and conquer,---------------------------------------------------------------------------------------------------------------------------------------- -----Date of Submission: 23, October, 2012 Date of Publication: 10, November 2012-------------------------------------------------------------------------------------------------------------------------------------------- I. Introduction For examp le, 4+4+4+4+4+4+4=28 is Multiplication is not new in the field of equivalent to the multip licat ion equation 7*4=28.mathematics. It was developed in ancient age.Around 18000BC it was developed [6]. II. Previous WorkMultiplication is the process of mult iplying or Before discussing the proposed process ofextending some values to a desired value. For this mu ltip licat ion let us take a look to those previousprocess we need two numbers: mu ltiplicand and works that are available at present. To make ourmu ltip lier. Mu ltiplicand is the number which will be presentation clear it is necessary to discuss ltip lied and multip lier is the number by whichmu ltip licand will be multip lied. We know that A. Repeated additi on multi plication: [4][5]mu ltip licat ion is commutative and distributed over Among the available mu ltiplication algorith msaddition and subtraction [6]. the most simple one is repeated addition In the field of mu ltip lication we notice many mu ltip licat ion. It has the complexity of O(n) all thenew developed processes, many new techniques are time. The logic behind this process is very simple,being proposed. Many of them are widely accepted. simp ly decrease the multiplier by 1 and add theThere are many variat ions in the technique of mu ltip licand to the accumulator and repeat themu ltip licat ion. We will see shortly many of these process until the mult iplier become 0(zero). It is cleartechniques in brief. that the process will run number of t imes equals to Let see the original process of the mult ltip licat ion. In the mu ltip lication process there aretwo main factors namely mu ltiplicand and multip lier. B. Shift and add [4][5 ]The operation of mult iplication is performed as a The next process of mult iplication is “shift andsummation of mu ltiple additions. The following add”. This process is performed on the binaryprocess describes it in short. The result of number. Here the LSB of multip lier is checked, if itmu ltip licat ion is the total number (product) that is 1 then mu ltiplicand is added otherwise the previouswould be obtained by combin ing several (mu ltip lier) partial product is shift one bit to the right. Thisgroups of similar size (mu ltip licand). The same result process is continued until the MSB of the multipliercan be obtained by repeated addition. If we are is checked. This is a sequential multiplication schemecombin ing 7 groups with 4 objects in each group, we with worst case time delay proportional to thecould arrive at the same answer by addition. numebr of bits O(n). So in this process we notice The IJES Page 18
  2. 2. Multiplication another Hue And Crythe pertial product of the process is less tahn the E. Booth’s Multi plication[1][3]previous process. But in this process if all the Booths mul ti plication algorithm is a multip licationmultiplier bit is 1 jthen the total number of pertial algorith m that multip lies two signed binary numbersproduct is equal to the number of bit present in the in twos co mplement notation. The algorith m wasmultiplier. invented by Andrew Donald Booth in 1951 wh ile doing research on crystallography at BirkbeckC. Array multi plier [4][5] College in Bloo msbury, London. Booth used desk The next topic is to be discussed is Array calculators that were faster at shift ing than addingmultiplier. Checking the bits of the multiplier one at and created the algorithm to increase their speed.time and forming the pertial product is a shift Booths algorithm is of interest in the study ofoperation that requirs a sequence of add and shift computer architecturemicroperation. The multiplication of two binarynumbers can be done with one micro operation bymeans of a combinational circuit that forms the Booths algorithm involves repeatedlyproduct bits all at once. adding one of two predetermined values Multiplicand and Multiplier to a product, then performing a Let us illustrate the process with an rightward arith metic shift on product. Let M and Qexample, let there are two bit multiplicand be the multiplicand and multip lier, respectively; and let x and y represent the number of bits in m and r. Then the number of bit in the product is equal to theThe operation is as follows (x+y+1). The Booth’s multip licat ion process is based on the four basic steps. In this process the two mu ltip lier bit is checked and depending on their combination (2^2=4) four different steps is performed. Now let us discuss the Booth’s mu ltip licat ion by taking an examp le. A combinational circuit binary multiplierwith more bits can be constructed in a similar Let M(mu lt iplicand) = 000000000001101fashion. A bit of the multiplier is ANDed with eachbit of the multiplicand in as many levels as there are Q(mu ltip lier) =0000000001010101bits in multiplier. Binary out put in each level ofAND gate is added in parallel with partial product of M =0000000000001101the product. For j multiplier bits and k multiplicandbits wee need j*k AND gates and (j-1)*k bits addersto produce a product of j+k bits. Q =0000000001010101 As our process is based on the concept of =1111111111110011 // mult iplier bit 10the Booth’s multip lication we shall discuss the =000000000001101* // mult iplier bit 01Booth’s process next . =11111111110011** // mult iplier bit 10 =0000000001101*** // mult iplier bit 01D. Divide and conquer [2] =111111110011**** // mult iplier bit 10 In traditional process of mu ltip licat ion we see that =00000001101***** // mult iplier bit 01the process of mu ltiplication of two n b it binary =1111110011****** // mult iplier bit 10number require O( ) digit operation . but in d ivide- =000001101******* // multip lier b it 01n-conquer technique the multip lication requires =0000010001010001 // productO( ) or O( ) (appro ximately) bit operation. In this process binary number is divided into two In the above examp le we notice that there are threeparts: say the number is X, so the number can be operations performed, namely shift, addition,represented as X= a + b. So in this process we see complement.that there may be four n/2 bit operation but the The number of operation is as followsproduct (say X*Y) can be represented as three n/2 bit Shift = 7mu ltip licat ion. So in this process the complexity can Addition= 7be represented as O( ) = O( ). Co mplement= The IJES Page 19
  3. 3. Multiplication another Hue And Cry I. MOTIVATION In the proposed process we have seen a new One of the techniques of mu ltip lication is Shift process of mu ltip licat ion. No w let us take an exampleand Add mult iplication. We observed the modified to illustrate the process and show how efficient thisShift and Add process by Feynman. This technique is process is. The example is given below.illustrated in below to clear the topic. Let M=10 and Q=10011 Let Mult iplicand(M) is 22(10110) and So according to the proposed process themu ltip lier(Q) is 5(101). So the multip lication process mu ltip licat ion of the numbers is as followsis as follo ws M =10110 M =00000010 Q =101 Q =00010011 10110 =11111110 //mult iplier b its 110 10110** =000010** // mult iplier bits 001 1101110 =1110**** //mult iplier b its 010 =010***** //mult iplier b its 001 In the above process we notice that the process =00100110 //final productof mult iplication is a Shift and Add multip licationprocess, but when a 0(zero) occur the shift as twice. So in the illustrated process we notice that a bigSo here is the logic that when there are one or more advantage is achieved. If the booth process is appliedthan one 0s we can take them as one sequence of in the same mu ltiplier (Q) the number of partial0s.We can apply this logic to some other algorithm to product, shift operation and complement operationmake that algorithm more efficient. are as follows Partial product =4 Shift operation =5 II. PROPOS ED PROCESS Co mplement operation=2 In the previously discussed process we notice Whereas the same operations in the proposed is asone thing that when there is a sequence of 0’s we followsconsider each 0(zero) for one single step. Here for Partial product =3this reason the number of steps increases each time. Shift operation =5So our observation is that if we can consider a Co mplement operation=2sequence of zeros in a single step then the whole In the Booth’s process when there is a longnumber of steps decreases. One more thing when sequence of 0’s or 1’s the whole sequence is treatedthere a sequence of 1’s we can take the sequence as individually, whereas in our p roposed the wholeone step. We notice this process can be applied to sequence is treated at a single step. As a result theBooth’s algorithm to make it more efficient. number of steps is decreased. It is easily The proposed algorithm is as follows understandable that if the numbers of steps decreaseAlgorithm: the space complexity also decreased. And also it is While starting this algorith m we will take noticeable that as the number of bits decreased theone thing in consideration that one extra 0 is added partial products also decrease. Here is the main goalas the LSB. Three bits will be considered at a time in of the proposed process.general and the last bit fro m previous bit pattern will If we consider chips to perform these operationsbe considered. then we can see that the time required for shiftStep 1. If the bits are 000 then continue checking operation (4 bit bidirectional Un iversal shift registerthe bits until the bit is 1. Then shift the partial LS 194) is 1/ 36 ns [7] [8], add operation (4 b it carryproduct to the right same as the number of the 0s. look ahead adder LS 7483) is 45 ns [7] [8] and forStep 2. If the bits are 001 then continue the complement operation (LS hex inverter 7404) is 12checking the next bit until it is 1. Then add ns [7] [8]. So here we can make an assumption thatmu ltip licand and shift the partial p roduct to the right how much time is saved if we apply this process ofas the number of 0s. mu ltip licat ion instead of others.Step 3. If the bits are 010 then subtract multiplicandand shift the partial product three positions right.Step 4. If the bits are 101 then add multiplicand andshift the partial product three positions right.Step 5. If the bits are 110 then check the next bitsuntil it is 0 and subtract the multiplicand and shift thepartial p roduct to the right as the number of The IJES Page 20
  4. 4. Multiplication another Hue And Cry III. Future Scope In the proposed process of multip licat ion wenotice that if we consider the consecutive 0’s or 1’sin a single step then the number of partial product Dr. Samar Sen S armadecreases also the other operations like shift Professor, Depart ment of Co mputeroperation and complement operation. So here the Science and Engineering, Un iversitytotal number of other operations also decreases and Of Calcutta, Indiahence the space and time co mplexity of the wholeoperation also decreases. We notice that the divideand conquer process can also decreases the time andspace required for an operation. So we will try tomerge the proposed process of mult iplication with thedivide and conquer technique so that the total numberof operation with the time and space comp lexity canbe decreased. We notice in different case wheredivide and conquer process technique were appliedand the result were as expected. We think in our caseit would not be an exception. IV. Conclusion In our proposed process we notice that sometime it seems that considering the sequence of 0’s and1’s at a single step may increase the maintenancecomplexity and over burden the whole process. Butwhen the proposed process is in real the process is abig advantage over the existing process ofmu ltip licat ion. REFERENCES[1] A. Booth, “A signed binary multiplication technique,” Q. J. Me& Appl. March., vol. 4, pp. 236-240, 19.51.[2] Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullmant “The design and analysis of computer algorithm” Pearson, 1974.[3] Booth Multiplication Algorithm Abenet Getahun Fall 2003 CSCI 401[4] John P. Hayes “Computer architecture and organization”, Second Edition, McGraw-Hill, 1988.[5] M. Morris Mano “Computer System Architecture”, Third Addition, Prentice Hall, 1993.[6], visited at 12 nov. 2012[7] William D.Anderson, Roberts Loyd Morris, John Miller “Designing with TTL integrated circuit”, McGraw-Hill, 1971[8] “TTL data book for design Engineer”, 2 nd edition, Texas instrument Inc, 1981 Barun Biswas CVPR Unit, Indian Statistical Institute, 203 B.T Road Ko lkata-700108, India Krishnendu Basuli Assistant Professor, Depart ment of Co mputer Science, West Bengal State The IJES Page 21