Implementation of 16 x16 bit multiplication algorithm by using vedic mathematics over booth algorithm

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 04 Issue: 05 | May-2015, Available @ http://www.ijret.org 371
IMPLEMENTATION OF 16x16 BIT MULTIPLICATION ALGORITHM
BY USING VEDIC MATHEMATICS OVER BOOTH ALGORITHM
Pranita Soni1
, Swapnil Kadam2
, Harish Dhurape3
, Nikhil Gulavani4
1
Department of E&TC, Sinhgad Academy of Engineering, Pune, Maharashtra, India
2
3
4
Abstract
Multiplication is one of the important operation in digital signal processors. The speed of processor depends on the hardware
architecture, delay and power. Previously implemented Booths algorithm is not very competent in terms of delay and hardware
complexity, therefore we have implemented multiplication algorithm which is efficient over booths algorithm. For multiplication,
as number of bit increases the respective delay increases. For efficient processors, delay should be minimum, to avoid increase in
delay we require minimum hardware architecture for the processor and Vedic multiplier has minimum hardware architecture.
In this paper we have explained 16 bit ‘Urdhva Tiryagbhyam’ Vedic Multiplier and Booth Multiplier and compared on the
various parameters such as speed, delay, hardware complexity. These algorithms are executed in VHDL language by using model
sim and synthesis is done in Xilinx software. Spartan 3 family FPGA development board is used for hardware implementation of
these algorithms.
Keywords: Urdhva Tiryagbhyam, Booth, Vedic Multiplier, Spartan 3.
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1. INTRODUCTION
Multiplication is one of the fundamental function in
arithmetic operations. In a Digital Signal Processors
multipliers are the key components. Since multiplication
dominates the execution time of many DSP algorithms, so
there is need of high speed multiplier. The demand of high
speed processing is increasing in computer and DSP
applications. Efficient arithmetic operations are important to
achieve the desired performance in many real-time signals.
Hence continuous efforts are taken to improve the
performance of multiplier by reducing the speed. There are
several methods have been used for multipliers such as
Braun-array, Baugh-Wooley method of two’s complement
and Booths algorithm. Booth’s algorithm is the most
successful method for multiplication.
In this paper a simple 16X16 bit multiplier is proposed
which is based on Urdhva Tiryagbhyam sutra of Vedic
mathematics and on Booth’s algorithm. The two numbers of
16 bit each are multiplied by using Vedic maths. The main
concept behind this is that to reduce the propagation delay
of the architecture which is the drawback of Booth’s
algorithm. As the number of bit increases in the Booth’s
algorithm delay increases noticeably and for Vedic
mathematics as number of bit increases delay increases
slightly. In the end we have compared the two algorithms on
the basis of certain parameters and proved that Vedic
mathematics is the best way to multiplication of higher
number of bits like 16X16 bit.
2. BOOTHS MULTIPLIER
The previous method for multiplier is the Booth’s multiplier
[5] based on Booth’s algorithm. This method is given by
“Andrew Donald Booth”. The given flowchart describes the
method.
Fig 1: Flow chart for Booths Algorithm

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The Booth’s multiplier scans the three bits at a time to
reduce the number of partial products. These three bits are
the two bits from the present pair and third bit from high
order bit of an adjacent lower order pair.
From above flow chart we have algorithm as follows:
1. The multiplier and multiplicand are placed in Q and B
registers respectively.
2. A one bit Q(-1) register is placed at the least significant
bit of Q(0) of register Q.
3. The final result appears in A.
4. A and Q(-1) registers are initialized to 0.
5. Multiplication of no is one in n cycle.
6. In each cycle Q(0) and Q(-1) are examined.
i. If Q(0) and Q(-1) are (1-1 or 0-0) then all the
bits of A ,Q and Q(-1) registers are shifted right
by 1 bit.
ii. If Q(0) and Q(-1) are 01 then multiplicand is
added with A. After addition A Q and Q(-1)
registers are shifted right by 1 bit.
iii. If Q(0) and Q(-1) are 10 then multiplicand is
subtracted from A . After subtraction A Q and
Q(-1) registers are shifted to right by 1-bit.
7. The final result appears in A.
In earlier days this Booth’s multiplier is very suitable for
multiplication. This will give the accurate result of
multiplication. But this multiplier has some drawback.
The drawbacks can be overcome by designing a suitable
multiplier for 16X16 bit multiplication. We will design the
new multiplier as Vedic multiplier by using the concept of
Vedic mathematics.
3. VEDIC ALGORITHM
Vedic multiplier is based on the Vedic mathematics [1].
Vedic mathematics is the ancient method of mathematics. It
was rediscovered from Vedas in between 1911 to 1918 by
Sri Bharti Krishna Tirthaji (1884-1960). Who was
mathematician and Sanskrit scholar. The Vedic mathematics
is based on the 16 sutras of Urdhva Tiryagbhyam. It simply
means that “vertically and crosswise multiplication”.it
involves minimum number of calculations, it reduces the
space, save the computational time and it is applicable in all
cases of multiplication. This method is most efficient when
the number of bit increases in multiplication.
The structure of this method is shown in the figure from this
we get clear idea of “vertical and crosswise multiplication”
Fig 2: Vedic method for multiplication of 4 bit binary
Number
For multiplication of two 4 bit numbers the multiplication
process is divided into 7 steps. The first number is a3a2a1a0
which is multiplier and second number is b3b2b1b0 which is
multiplicand as shown in the step1 of the figure. The LSB
(least significant bit) of multiplier is multiplied with LSB of
the multiplicand and the result of this multiplication is
stored as the LSB of the final result. Then as shown in the
step2 of the above figure The LSB of multiplier is
multiplied with second higher bit of multiplicand and
second higher bit of multiplier is multiplied with LSB of
multiplicand and then these two partial products are added,
after adding these two numbers the LSB of addition is taken
as the second higher bit of the final result and remaining bits
of the addition are taken as carry bit and this carry can be of
multi-bit. Then follow the steps which are given in the
above figure and after that we get the result of the 4 bit
multiplication.
In this paper we have done 16 bit multiplication and for that
we require different modules of the 2 bit, 4 bit, 8 bit Vedic
multiplier. The different modules of multiplier and their
architectures are explained below.
4. THE PROPOSED MULTIPLIER
ARCHITECTURE
This method can be used for any N X N bit multiplication
and this Vedic multiplier is independent on clock frequency
because partial product and their sum calculated in parallel.
And because of we don’t require high clock frequencies for
multiplication and that’s why less switching takes place,
because of less switching delay and power minimizes and
the result of this we get an efficient processor in delay and
power. In this paper we are presenting 16 bit multiplication
and for that we require 2 bit, 4 bit, 8 bit multiplier. The
detail explanation of these architectures is given below.

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4.1 Vedic Multiplier for 2X2 Bit Module
Fig 3: Vedic multiplier for 2X2 bit
A basic block diagram of 2X2 multiplier is as shown. In this
diagram a0, a1 are the bits of first digit and b0, b1 are the
bits of second digit. In this multiplier we are taking AND of
respective bits and then as shown in block diagram we are
following the procedure. The multiplication is done on the
sutra of Urdhva-Tiryagbhyam. The result obtained is of 4
bits.
After implementation of basic 2X2 multiplier it is easy to
implement 4X4 multiplier. In the diagram a0, a1, a2, a3 are
the bits of first digit while b0, b1, b2, b3 are the bits of
second digit. For 4X4 bit multiplication we require four 2X2
multiplier, three adders. After doing the procedure as shown
in figure 4 we can implement the 4X4 Vedic multiplier. The
result obtain is of 8 bits.
This multiplier is implemented from the previous 4X4
multiplier. In this diagram a0 to a7 are the bits of first digit
and b0 to b7 are the bits of second digit. For 8X8 multiplier
we require four 4X4 multiplier and three 8 bit ripple carry
adder after doing the procedure as shown in figure 5 we will
get the result of 8X8 multiplier and the result obtained is of
16 bits.
This is the last module of the Vedic multiplier, this
multiplier is implemented from the 8X8 multiplier. In this
diagram a0 to a15 are the bits of first digit and b0 to b15 are
the bits of second digit. For 16X16 multiplier we require
four 8X8 multiplier and three adders after doing the
procedure as shown in figure 6, we will get the result of
16X16 multiplier and the result obtained is of 32 bits.

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5. IMPLEMENTATION ON XILINX SOFTWARE
Now we are moving towards main programming part. Using
the Vedic Multiplier we have to develop a program. In this
work the 16X16 bit multiplier is designed in VHDL (very
high speed integrated circuits hardware description
language). Synthesis and simulation was done in XILINX
ISE 10.1. [6] Project navigator and simulator integrated in
the XILINX package. XILINX is software which is purely
based on the VHDL language. VHDL is Very large scale
integrated circuit Hardware Description Language. We have
performed our programming using the Spartan-3 family,
Package is selected as PQ208 and Speed grade is -5.
Integrated Software Environment is a software tool
produced by Xilinx for synthesis and analysis of HDL
design, enabling the developer to synthesize (compile) their
design, perform time analysis, examine RTL diagrams,
simulate a design reaction to different stimuli, and configure
the target device with the programmer. It includes ISE
simulator which we can use for functionality verification of
program.
The programming is divided into 4 parts. For developing the
multiplier of 16X16 bit, we have to first develop the
multiplier of 2X2, 4X4 and 8X8.
6. HARDWARE IMPLEMENTATION
After the programming is done in the XILINX software the
hardware part comes into account. To see easily the output
of multiplication we have executed the output on FPGA
(Field Programmable Gate Array) kit. The kit is having
many LED’s which we are using for input and output
purpose.
For downloading the program we require to join the JTAG
(Joint Test Action Group) cable. One port is attached to the
kit and the other port is attached to the CPU. After the
program is downloaded we will see different combinations
of 16X16 bit multiplication.
The use of this FPGA kit is that the user will see the output
easily.
7. RESULTS
The result of the 16X16 Vedic multiplier is given in the
figure’s below in which 16 bit number is in binary and
decimal form. Also the RTL schematic of Vedic and Booth
multiplier is given.
Fig 7: Result of 16X16 multiplier in binary form.

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Fig 8: Result of 16X16 multiplier in decimal form.
Fig 9: RTL schematic of 16X16 Booth’s multiplier. Fig 10: RTL schematic of 16X16 Vedic multiplier.

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From the above RTL schematic it is clear that the hardware
requirement for the Booths algorithm is much more than the
Vedic multiplier, and that’s why the hardware complexity of
Vedic algorithm is less than the Booth’s multiplier.
8. COMPARISON OF DEVICE UTILIZATION
BETWEEN BOOTH’S AND VEDIC
MULTIPLIER
We have compared Booth’s and Vedic multiplier on the
basis of device utilization which is as given below.
8.1 4x4 Booth’s and Vedic Multiplier
Table 1: Comparison of device utilization theory of 4X4
multiplier
Parameter Booth’s
multiplier
Vedic multiplier
Combination
path delay
13.224 ns 13.363 ns
No. of slices 23 out of 3584
(0%)
18 out of 3584
(0%)
No of 4 input
LUTs
42 out of 7168
(0%)
32 out of 7168
(0%)
No of bonded
IOBs
16 out of 141
(11%)
16 out of 141
(11%)
8.2 8x8 Booth’s and Vedic Multiplier
multiplier
Parameter Booth’s
multiplier
Vedic multiplier
Combination
path delay
50.866 ns 21.782 ns
No. of slices 185 out of
3584 (5%)
75 out of 3584
(2%)
No of 4 input
LUTs
330 out of
7168 (4%)
135 out of 7168
(1%)
No of bonded
IOBs
32 out of 141
(22%)
32 out of 141
(22%)
8.3 16x16 Booth’s And Vedic Multiplier
multiplier
Parameter Booth’s
multiplier
Vedic
multiplier
Combination
path delay
93.86 ns 28.316 ns
No. of slices 718 out of 3584
(20%)
345 out of 3584
(9%)
No of 4 input
LUTs
1281 out of
7168 (17%)
622 out of 7168
(8%)
No of bonded
IOBs
64 out of 141
(45%)
64 out of 141
(45%)
8.4 Comparison of Combination Path Delay
between Booth’s and Vedic Multiplier
Fig 11: comparison of combination path delay between
booth’s and Vedic multiplier
In the figure 9 the comparison of combinational path delay
between 4,8,16 bit Booth and Vedic multiplier is given.
From that we can clearly conclude that the efficiency of
Vedic multiplier is greater than Booth’s multiplier. As no of
bit increases booth become more efficient. Hence for that
reason it is very important to use Vedic multiplier instead of
Booth’s multiplier.
9. CONCLUSION
In this paper we have developed the suitable architecture for
multiplication using Vedic multiplier. This multiplier is
suitable for multiplication of large number of bits. We
compared Vedic multiplier over Booths multiplier and
successfully show that Vedic multiplier is more convenient
than Booths multiplier. The simulation results clearly shows
that the Vedic multiplier is having minimum path delay as
compared to the Booth’s multiplier also hardware
complexity of Vedic algorithm is less than the Booth’s
algorithm.
REFERENCES
[1]. Jagadguru Swami Sri Bharati Krisna Tirthaji Maharaja,
“Vedic Mathematics: Sixteen Simple Mathematical
Formulae from the Veda,” Motilal Banarasidas Publishers,
Delhi, 2009, pp. 5-45
[2]. Virendra Babanrao Magar, “Intelligent and Superior
Vedic Multiplier for FPGA Based Arithmetic Circuits”,
International Journal of Soft Computing and Engineering
(IJSCE) ISSN: 2231-2307, Volume-3, Issue-3, July 2013.
[3]. Perry, “VHDL”, McGraw Hill Publication.
[4]. VHDL Primer by J.Bhaskar.
[5]. A. D. Booth, “A signed binary multiplication
technique,” Q. J. Mech.Appl. Math., vol. 4, pp. 236–240,
1951.
[6]. Xilinx ISE User manual.
[7] .S.S.Kerur, Prakash Narchi, Jayashree C N, Harish M
Kittur and Girish V A “Implementation of Vedic Multiplier
for Digital Signal Processing” International conference on
VLSI communication & instrumentation (ICVCI), 2011.

Implementation of 16 x16 bit multiplication algorithm by using vedic mathematics over booth algorithm

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Implementation of 16 x16 bit multiplication algorithm by using vedic mathematics over booth algorithm