Analysis of different multiplication algorithm and FPGA implementation of recursive barrel shifter method for multiplication
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The document describes a new recursive barrel shifter algorithm for multiplication and its FPGA implementation. It analyzes existing multiplication algorithms like Booth, Wallace tree, Karatsuba and their limitations. The proposed algorithm uses a barrel shifter recursively by counting the number of ones in each number, shifting the multiplicand by powers of two, and adding the results. This is implemented on a Xilinx FPGA for performance analysis in terms of speed, power and area.
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072
© 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1141
Analysis of different multiplication algorithm and FPGA
implementation of recursive barrel shifter method for multiplication
Manoj M Kamble1, Dr. Sunita P Ugale2
1 Student, Dept of E&TC, K.K.Wagh IEER, Maharashtra, India.
2 Associate Professor, Dept of E&TC, K.K.Wagh IEER, Maharashtra, India.
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - Many of the today’s real time signal
processing algorithm included multiplication as its
processing heart. In case of signal and image
processing, it mostly used functional unit. In this review
paper we are introducing logically new, fast and more
efficient multiplication algorithm that seem to fulfilled
fast processing requirement. This paper introduces how
effectively we can use barrel shifter for implementation
of multiplication method. Further work will carried
comparative study of different multiplier with respect
to some parameters like cost, power consumption, area
and speed. For implementation and parametric
analysis, we are using XC3S400 FPGA hardware
platform, VHDL coding language for hardware
description. Xilinx ISE-simulation tool has many inbuilt
compatible tools for parameter analysis like XPE for
power estimation that we are using here. Paper
comprises description of current scenario, existed
multiplication methods and our proposed algorithm.
Key Words: recursive barrel shifter, Karatsuba, binary
one counter, Power predictor (PP), Vedic math.
1. INTRODUCTION
Now days, for many application specific processor
implementations such as real time signal processing,
image processing, encryption, data manipulation requires
high speed integrated circuit. Many signal processing
algorithm make use of math-multiplier. There are different
multiplication algorithm presently available and have
implemented on SOC. Booth multiplier, Wallace tree
multiplier, Karatsuba multiplier, Braun algorithm are
more popular multiplication algorithm [2]. But many of
them having their own restriction in terms of speed, on
chip area, use of logic gates, energy, cost per cell, so in our
project we are analyzing these many parameters and
going to introduced one modified multiplication algorithm
based on barrel shifter.
Barrel shifter has ability to shift any number to left or
right (as per requirement) by N-position within only one
clock cycle [1]. If we analyze this convenient barrel shifter
behavior can used to implement multiplier that can
multiply multiplicand number with multiplier number
which should be perfect representation in power of 2 [3].
We are proposing new method based on barrel shifter
used for multiplication of numbers which are not perfect
two power representation.
1.1 Overview of existed algorithms
Multiplication methods that many of processors are using
today are inspired by Vedic multiplier introduced in
Indian Vedas with different 16 sutras. Vedic multiplier
uses bit-wise multiplication with simultaneous product
term finding and it’s column-wise addition. It is one of the
best benchmark for fast multiplication algorithm. Vedic
math have introduces commonly used two methods as
vertical crosswise multiplication and Nikhilam method [4].
Array multiplier is one of the bit-by-bit multiplication
implementation approaches. Partial products are first
generated and stored in memory from where it given to
array of summation. As it performs operation on data in
array form, so dedicated memory space is first prior to its
implementation. Broun has described array multiplication
with carry propagation adder [2].
Shift and add algorithm takes use of serial shifter and
parallel adder. Both multiplicand and multiplier have
dedicated or overlapped memory locations. This has
improved at many stages to decreases size required to
store multiplicand, multiplier and result. It checks for each
bit of multiplier and take decision about what to do with
multiplicand [2].
Another fast and more popular multiplication algorithm is
Karatsuba algorithm which was named after Russian
mathematician Anatolii Alexeevitch Karatsuba. Basic idea
was to split the numbers into its two halves and find three
auxiliary product terms u, v and w as below
X = PQ = 10*P+Q
Y = RS = 10*R+S,](https://image.slidesharecdn.com/irjet-v3i1198-171023060231/85/Analysis-of-different-multiplication-algorithm-and-FPGA-implementation-of-recursive-barrel-shifter-method-for-multiplication-1-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072
© 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1142
Result = X*Y
= (10*P+Q) * (10*R+S)
= (P*R)*100 + ((Q*R) + (P*S))*10 + (Q*S)
= U*100 + middle*10 +W
V = (Q-P) * (S-R)
Middle term = U + W – V = (Q*R) + (P*S)
Thus, for N-bit number multiplication, the above same
process can be use with multiple repetitions, but N should
be the perfect representation in power of two [5].
Many of the above discussed multiplication algorithms
seems to be reducing addition array size, but Wallace tree
multiplier having ability to reduce number of total partial
product terms as well. It make use of Booth Recode radix-x
(x = 2, 4, 8) method for generating minimum partial
product terms and 3:2, 4:2, 5:2 compressors to reduce
addition further. It makes use of carry save adder at initial
stages and carry propagation adder at final stage [6].
2. PROPOSED ALGORITHM
Here our algorithm makes use of barrel shifter; recursively
for final computation. As we depicted above that barrel
shifter can effectively used for number multiplication
those are perfectly represented in power of two. But it can
be used recursively if stated condition is not satisfied. Just
below we have listed step to be followed by our algorithm-
1) Suppose X and Y are two each of N-bit numbers.
2) Count the number of ones in each number.
3) Number with smaller ones count will treat as a
multiplier and other will be multiplicand.
4) Finding further successive power two
representations of multiplier.
5) Now shift multiplicand by power of two to left
6) Give shifted result to adder
7) Repeat step 4 to 7 till power of two become zero.
8) Final adder result will be multiplication result
In step 3, selecting less ones count number as a multiplier
so as to avoid unwanted shifting operation per clock cycle.
Algorithm explanation with example gives as below-
Let us consider two number as A=11 (bX1011) and
B=9(bX1001), ones count as OA(3) and OB(2)
B having two power indexes as +
Shifting of A by 3 = (bX01011000)…… (1)
Shifting of A by 0 = (bX00001011)…… (2)
Final result will be as = (1) + (2) = (bX01100011) = 99.
Flowchart of algorithm gives clear idea of operation
sequence-
Fig 1: Algorithmic flowchart
3. CONCLUSIONS
Thus we have studied different existed multiplication
algorithms and it restrictions in terms of power
consumption, cost and speed. We have depicted in our
algorithm the effective use of barrel shifter for
multiplication.
ACKNOWLEDGEMENT
The authors would like to acknowledge the support of
K.K.Wagh IEER, Nasik for providing the research facilities.
REFERENCES
[1] M. Seckora, “Barrel Shifter or Multiply/Divide IC
Structure,” U.S. Patent 5,465,222, November 1995.
[2] Cem Ergun, seminar at Eastern Mediterranean
University on ‘Multiplication & Division Algorithms’.
[3] P. Saha, A. Banerjee, P. Bhattacharyya, A. Dandapat,
2011 ‘High Speed ASIC Design of Complex Multiplier Using
Vedic Mathematics’ IEEE Students' Technology Symp-
osium.
[4] Mrs.Toni J.Billore, Prof.D.R.Rotake May-Jun. 2014
‘FPGA implementation of high speed 8 bit Vedic Multiplier
using Fast adders’ IOSR Journal of VLSI and Signal
Processing, PP 54-59.
[5] A. A. Karatsuba: The Complexity of Computations.
Proceedings of the Steklov Institute of Mathematics,
http://www.ccas.ru/personal/karatsuba/divcen.pdf.
[6] H. Bansal, K. G. Sharma, T. Sharma ‘Wallace Tree
Multiplier Designs: A Performance Comparison Review’
Innovative Systems Design and Engineering, 2014.
LargerSmaller
Ones count block
Comp.
Multiplier
Register
Multiplicand
Register
X Y
Barrel shifter
Adder
RESULT
PP](https://image.slidesharecdn.com/irjet-v3i1198-171023060231/85/Analysis-of-different-multiplication-algorithm-and-FPGA-implementation-of-recursive-barrel-shifter-method-for-multiplication-2-320.jpg)
Analysis of different multiplication algorithm and FPGA implementation of recursive barrel shifter method for multiplication
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 © 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1141 Analysis of different multiplication algorithm and FPGA implementation of recursive barrel shifter method for multiplication Manoj M Kamble1, Dr. Sunita P Ugale2 1 Student, Dept of E&TC, K.K.Wagh IEER, Maharashtra, India. 2 Associate Professor, Dept of E&TC, K.K.Wagh IEER, Maharashtra, India. ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - Many of the today’s real time signal processing algorithm included multiplication as its processing heart. In case of signal and image processing, it mostly used functional unit. In this review paper we are introducing logically new, fast and more efficient multiplication algorithm that seem to fulfilled fast processing requirement. This paper introduces how effectively we can use barrel shifter for implementation of multiplication method. Further work will carried comparative study of different multiplier with respect to some parameters like cost, power consumption, area and speed. For implementation and parametric analysis, we are using XC3S400 FPGA hardware platform, VHDL coding language for hardware description. Xilinx ISE-simulation tool has many inbuilt compatible tools for parameter analysis like XPE for power estimation that we are using here. Paper comprises description of current scenario, existed multiplication methods and our proposed algorithm. Key Words: recursive barrel shifter, Karatsuba, binary one counter, Power predictor (PP), Vedic math. 1. INTRODUCTION Now days, for many application specific processor implementations such as real time signal processing, image processing, encryption, data manipulation requires high speed integrated circuit. Many signal processing algorithm make use of math-multiplier. There are different multiplication algorithm presently available and have implemented on SOC. Booth multiplier, Wallace tree multiplier, Karatsuba multiplier, Braun algorithm are more popular multiplication algorithm [2]. But many of them having their own restriction in terms of speed, on chip area, use of logic gates, energy, cost per cell, so in our project we are analyzing these many parameters and going to introduced one modified multiplication algorithm based on barrel shifter. Barrel shifter has ability to shift any number to left or right (as per requirement) by N-position within only one clock cycle [1]. If we analyze this convenient barrel shifter behavior can used to implement multiplier that can multiply multiplicand number with multiplier number which should be perfect representation in power of 2 [3]. We are proposing new method based on barrel shifter used for multiplication of numbers which are not perfect two power representation. 1.1 Overview of existed algorithms Multiplication methods that many of processors are using today are inspired by Vedic multiplier introduced in Indian Vedas with different 16 sutras. Vedic multiplier uses bit-wise multiplication with simultaneous product term finding and it’s column-wise addition. It is one of the best benchmark for fast multiplication algorithm. Vedic math have introduces commonly used two methods as vertical crosswise multiplication and Nikhilam method [4]. Array multiplier is one of the bit-by-bit multiplication implementation approaches. Partial products are first generated and stored in memory from where it given to array of summation. As it performs operation on data in array form, so dedicated memory space is first prior to its implementation. Broun has described array multiplication with carry propagation adder [2]. Shift and add algorithm takes use of serial shifter and parallel adder. Both multiplicand and multiplier have dedicated or overlapped memory locations. This has improved at many stages to decreases size required to store multiplicand, multiplier and result. It checks for each bit of multiplier and take decision about what to do with multiplicand [2]. Another fast and more popular multiplication algorithm is Karatsuba algorithm which was named after Russian mathematician Anatolii Alexeevitch Karatsuba. Basic idea was to split the numbers into its two halves and find three auxiliary product terms u, v and w as below X = PQ = 10*P+Q Y = RS = 10*R+S,
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072 © 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1142 Result = X*Y = (10*P+Q) * (10*R+S) = (P*R)*100 + ((Q*R) + (P*S))*10 + (Q*S) = U*100 + middle*10 +W V = (Q-P) * (S-R) Middle term = U + W – V = (Q*R) + (P*S) Thus, for N-bit number multiplication, the above same process can be use with multiple repetitions, but N should be the perfect representation in power of two [5]. Many of the above discussed multiplication algorithms seems to be reducing addition array size, but Wallace tree multiplier having ability to reduce number of total partial product terms as well. It make use of Booth Recode radix-x (x = 2, 4, 8) method for generating minimum partial product terms and 3:2, 4:2, 5:2 compressors to reduce addition further. It makes use of carry save adder at initial stages and carry propagation adder at final stage [6]. 2. PROPOSED ALGORITHM Here our algorithm makes use of barrel shifter; recursively for final computation. As we depicted above that barrel shifter can effectively used for number multiplication those are perfectly represented in power of two. But it can be used recursively if stated condition is not satisfied. Just below we have listed step to be followed by our algorithm- 1) Suppose X and Y are two each of N-bit numbers. 2) Count the number of ones in each number. 3) Number with smaller ones count will treat as a multiplier and other will be multiplicand. 4) Finding further successive power two representations of multiplier. 5) Now shift multiplicand by power of two to left 6) Give shifted result to adder 7) Repeat step 4 to 7 till power of two become zero. 8) Final adder result will be multiplication result In step 3, selecting less ones count number as a multiplier so as to avoid unwanted shifting operation per clock cycle. Algorithm explanation with example gives as below- Let us consider two number as A=11 (bX1011) and B=9(bX1001), ones count as OA(3) and OB(2) B having two power indexes as + Shifting of A by 3 = (bX01011000)…… (1) Shifting of A by 0 = (bX00001011)…… (2) Final result will be as = (1) + (2) = (bX01100011) = 99. Flowchart of algorithm gives clear idea of operation sequence- Fig 1: Algorithmic flowchart 3. CONCLUSIONS Thus we have studied different existed multiplication algorithms and it restrictions in terms of power consumption, cost and speed. We have depicted in our algorithm the effective use of barrel shifter for multiplication. ACKNOWLEDGEMENT The authors would like to acknowledge the support of K.K.Wagh IEER, Nasik for providing the research facilities. REFERENCES [1] M. Seckora, “Barrel Shifter or Multiply/Divide IC Structure,” U.S. Patent 5,465,222, November 1995. [2] Cem Ergun, seminar at Eastern Mediterranean University on ‘Multiplication & Division Algorithms’. [3] P. Saha, A. Banerjee, P. Bhattacharyya, A. Dandapat, 2011 ‘High Speed ASIC Design of Complex Multiplier Using Vedic Mathematics’ IEEE Students' Technology Symp- osium. [4] Mrs.Toni J.Billore, Prof.D.R.Rotake May-Jun. 2014 ‘FPGA implementation of high speed 8 bit Vedic Multiplier using Fast adders’ IOSR Journal of VLSI and Signal Processing, PP 54-59. [5] A. A. Karatsuba: The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics, http://www.ccas.ru/personal/karatsuba/divcen.pdf. [6] H. Bansal, K. G. Sharma, T. Sharma ‘Wallace Tree Multiplier Designs: A Performance Comparison Review’ Innovative Systems Design and Engineering, 2014. LargerSmaller Ones count block Comp. Multiplier Register Multiplicand Register X Y Barrel shifter Adder RESULT PP