IRJET-Design and Implementation of LNS based Approximate Multiplier using Mitchell’s Algorithm and Operand Decomposition

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1760
Design and Implementation of LNS based Approximate Multiplier using
Mitchell’s Algorithm and Operand Decomposition
G. Mohanapriya1, M. Nagiga Nasirin2, Mrs. J. Mary Suji Mol3
1,2Department of Electronics and Communication Engineering, Jeppiaar SRR Engineering College,
Tamil Nadu, India
3Assistant professor, Dept. of Electronics and Communication Engineering, Jeppiaar SRR Engineering College,
Tamil Nadu, India
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Abstract - In this project we study the approximate
logarithmic multiplier which is implemented via low power. It
is mainly used to improve the consumption of convolutional
neural networks for image classification. Themainadvantage
of logarithmic multiplier is intrinsic tolerance to error. It
converts multiplications into additions. Hence this paper
shows improvement in accuracy and efficient logarithmic
approach.
Key Words: Convolutional Neural Networks, Logarithm,
Antilogarithm
1. INTRODUCTION
Mitchell algorithm (MA) is used along with Operand
Decomposition (OD) in order to increase the accuracy and
decrease the error percentage. First the inputs are divided
into four using some basic logic operations. This is the main
reason for the increase in accuracy of the existing system
compared to Operand Decomposition. This methoddoes not
give the accurate output but gives the nearest result value
compared to Mitchell Algorithm alone. Hence itisverymuch
useful for applications like Convolutional Neural Networks
(CNN), embedded systems, Digital Image Processing (DIP),
Digital Signal Processing (DSP) and datacenters.
These applications needs low time delays and they
don’t need accurate exact values. So, Mitchell Algorithm is
used here but near accurate values are needed in some
applications of digital image processing. Hence Operand
Decomposition adds some accuracy tothesameapplications
by combining with Mitchell Algorithm. Mitchell Algorithm
and Mitchell Algorithm with Operand Decomposition
involves the same process but few steps may vary due to the
increase in accuracy. The comparison ofMAandMAwith OD
is clearly revealed in this project.
1.1 LOGARTHMIC NUMBER SYSTEM
Many applications like Digital Image Processing (DIP),
Digital Signal Processing requires multiplication for
calculations.Fixedpointnumbersystemsgivesmultiplication
but with circuit complexity. Logarithmic Number System
(LNS) will overcome this circuit complexity by introducing
some errors. These errors will makes the results less
accurate.
OperandDecompositionwillincreasetheaccuracylevelof
results by performing logical operations of the input before
performing the algorithm.
1.2 MITCHELL ALGORITHM
Mitchell algorithm is done by taking logarithmforthetwo
binary inputs, adding them and finally taking antilogarithm
for the added result. Thus, the multiplication is converted to
addition.
2. DESIGN AND IMPLEMENTATION
Fig -1: OD block
DESCRIPTION-Two inputs with n numbers of bits are taken
and directed towards OD block for and the two inputs are
divided into four for further process.
LOGARITHM-Here multiplication is the main process.
Logarithm is used for convertingmultiplicationintoaddition
so the hardware complexity is being reduced.
ADDER-Adder is used to add the input values. Two types of
adder is used here, namely half adder and full adder.

ZERO DETECTOR-Zero detector is used to detect whether
the inputs are zero.
ANTILOGARITHM-After applying MA the result is
concatenated with 1 and number of zeros isappended based
on MSB position.
PROCESS
Two binary inputs X and Y are given in binaryform. Xand
Y are divided into A, B, C, and D. Taking MA separately for A,
B and C, D. LOD detects the leading bit position.Iftheleading
‘1’ is in 7th position the inputs are directly entered to the
barrel shifter else ENC and NOT gate will decide the shifting
count. L-Barr shift will then do the left shifting operation.
Adder will do the further adding operation of the inputs. Is
Zero block is used to check the two input values. So, if any
one of the input is zero it will directly give the result as zero
instead of doing all the operation. Here multiplication is
converted as addition. The results are added and
concatenated with 1 and number of zeros isappended based
on MSB position.
PROCEDURE
Step1: X, Y: n-bit binary multiplicands, OP=0:2n-bits
approximate product.
Step 2: Calculate A, B, C, D value using X and Y.
Step 3: Calculate A using the equation A=X|Y.
Step 4: Calculate B using the equation B=X&Y.
Step 5: Calculate C using the equation C= (~X) &Y.
Step 6: Calculate D using the equation D=X& (~Y).
Step 7: To take MA for A and B.
 A, B: n-bit binary multiplicands, OP1= 0:
2n-bits approximate product.
 Determine K1, leading ‘1’ position of 1st
number, A
number, B
 Evaluate X1 by shifting A by N-K1 bits
towards left
 Evaluate X2 by shifting B by N-K2 bits
towards left
 Calculate K12=K1+K2
 Calculate X12=X1+X2
 Decode K12 and insert ‘1’ in that position
of OP1
 Append X12 immediately after this one in
OP1
 A.B= OP1
Step 8: To take MA for C and D.
 C, D: n-bit binary multiplicands, OP2= 0:
2n-bits approximate product.
number, C
number, D
 Evaluate X1 by shifting C by N-K1 bits
towards left
 Evaluate X2 by shifting D by N-K2 bits
towards left
 Calculate K12=K1+K2
 Calculate X12=X1+X2
 Decode K12 and insert ‘1’ in that position
of OP2
 Append X12 immediately after this one in
OP2
 C.D= OP2
Step 9: Adding OP1 and OP2 we get OP
 OP=OP1+OP2.
Fig -2: Flowchart
ERROR PERCENTAGE-It can be obtained by the following
formula
Error %= ((Original value-Obtained value)/Original
value)*100
EXAMPLE
Step 1: Inputs
X=10001100(140), Y=00100101(37).
Step 2: Calculate A, B, C, D value using X and Y.

Step 3: A=X|Y
A= (10001100) | (00100101)
A=10101101
Step 4: B=X&Y
B= (10001100) & (00100101)
B=00000100
Step 5: C= (~X) &Y
C= (01110011) & (00100101)
C=00100001
Step 6: D=X& (~Y)
D= (10001100) & (11011010)
D=10001000
Step 7: Take MA for A and B
 Inputs
A=10101101,
B=00000100.
 MSB position of ‘1’ for K1 in binary form
MSB of A is ‘7’
K1=111
 MSB position of ‘1’ for K2 in binary form
MSB of B is ‘2’
K2=010
 After left shifting A
MSB of A is 7. So left shift by 0 (~ (K1))
X1=0101101
 After left shifting B
MSB of B is 2. So left shift by 5 (~ (K2))
X2=0000000.
 Adding K1 and K2
K12=K1+K2
K12=111+010=1001.
 Adding X1 and X2
X12=X1+X2
X12=0101101+0000000
X12=0101101.
 Concatenate 1 an X12
OP1= (1, X12)
OP1=1010110100
Step 8: Take MA for C and D
 Inputs
C=00100001,
D=10001000.
 MSB position of ‘1’ for K1in binary form
MSB of C is ‘5’
K1=101
 MSB position of ‘1’ for K2in binary form
MSB of D is ‘7’
K2=111
 After left shifting C
MSB of C is 5. So left shift by 2 (~ (K1))
X1=00001
 After left shifting D
MSB of D is 7. So left shift by 0 (~ (K2))
X2=00010
 Adding K1 and K2
K12=K1+K2
K12=101+111=1001.
 Adding X1 and X2
X12=X1+X2
X12=00001+00010
X12=00011
 Concatenate 1 an X12

OP2= (1, X12)
OP2=1000110000000
Step 9: Adding OP1 and OP2
OP=OP1+OP2
OP=0000001010110100+001000110000000
OP=0010110000110100(5172)
Error %= ((5180-5172)/5180)*100
Error %= 0.15
Table-1: Comparison between Mitchell Algorithm and
Operand Decomposition
X Y MA ERROR% OD ERROR%
140 37 1.15 0.15
117 157 5.92 1.57
203 183 10.41 0.76
The above table shows the comparison of errors between
Mitchell Algorithm and Operand Decomposition. From this
we came to know the efficiency of Operand Decomposition.
SOFTWARE REQUIREMENTS
Synthesis tool- Xinlinx ISE 14.5.
Verification tool-ModelSim 6.4c.
SNAPSHOTS
Picture tells everything. The below snapshots are very
much helpful for us in understanding about this paper. The
snapshots of output waveform, Look Up Tables (LUT) and
blocks are listed below.
Fig -3: Output waveform
Fig -4: Block diagram
This output and the block diagram clearly shows the two
inputs and the output.
The above diagram shows the several blocks insidethemain
block which are responsible several operations.
Fig-5 LUT
APPLICATIONS
 Digital Image Processing.
 Digital Signal Processing.
 Embedded systems and data centres.
3. CONCLUSION
From this we canclearlyunderstandaboutmultipliers. By
reading this report it is also easy to analyze and choose the
correct method for applications. If we need speedwithsome
accuracy Mitchell Algorithm along with Operand
Decomposition is the right technique to choose.

REFERENCES
[1] Z. Babic’, A. Avramovic’, P. Bulic’ (2010) An iterative
logarithmic multiplier.
[2] DurgeshNandan, JitendraKanungo, Anurag Mahajan
(2017) An efficient VLSI architecture for Iterative
Logarithmic Multiplier.
[3] John n. Mitchell, jr. t associate, ire (1962) Computer
Multiplication and Division Using Binary Logarithms.
[4] UrosLoric, PatricioBulic (2012) Applicability of
approximate multipliers in hardware neural networks.
[5] ZdenkaBabic,AleksejAvramovic,PatricioBulic (2008)An
iterative Mitchell’s Algorithm Based Multiplier.

IRJET-Design and Implementation of LNS based Approximate Multiplier using Mitchell’s Algorithm and Operand Decomposition

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