FPGA Implementation of High Speed 8bit Vedic Multiplier using Barrel Shifter

IDL - International Digital Library Of
Technology & Research
Volume 1, Issue 2, Mar 2017 Available at: www.dbpublications.org
International e-Journal For Technology And Research-2017
IDL - International Digital Library 1 | P a g e Copyright@IDL-2017
FPGA Implementation of High Speed 8bit Vedic
Multiplier using Barrel Shifter
Sahana Raj B S
Dept. of ECE, PESCE
Mandya, India.
sahanagandhi@gmail.com
Punith Kumar M B
Dept. of ECE, PESCE
Mandya, India.
punithpes@gmail.com
Abstract— In today’s world Vedic mathematics has
proved to be the most robust technique for arithmetic
operations. In contrast, conventional techniques for
multiplication provide significant amount of delay in hardware
implementation of n-bit multiplier. Moreover, the
combinational delay of the design degrades the performance
of the multiplier. Hardware-based multiplication mainly
depends upon architecture selection in FPGA or ASIC.
A barrel shifter is a digital circuit that can shift a data word by
a specified number of bits in one clock cycle. It can be
implemented as a sequence of multiplexers (mux.), and in
such an implementation the output of one mux is connected to
the input of the next mux in a way that depends on the shift
distance.
Keywords- Vedic Mathematics, Barrel Shifter, FPGA, Xilinx.
I. INTRUDUCTION
Arithmetic operations such as addition, subtraction
and multiplication are deployed in various digital circuits to
speed up the process of computation. Arithmetic logic unit is
also implemented in various processor architectures like
RISC, CISC etc. Arithmetic operations unit is a fundamental
building block of the central processing unit CPU) of a
computer, and even the simplest microprocessors contain one
for purposes such as maintaining timers. The processors found
inside modern CPUs and graphics processing units (GPUs)
accommodate very powerful and very complex Arithmetic
operations unit; a single component may contain a number of
Arithmetic operations unit. In general, arithmetic operations
are performed using the packed-decimal format. This means
that the fields are first converted to packed-decimal format
prior to performing the arithmetic operation, and then
converted back to their specified format (if necessary) prior to
placing the result in the result field.
Multiplication is an important fundamental function
in arithmetic operations. Multiplication-based operations such
as Multiply and Accumulate(MAC) and inner product are
among some of the frequently used Computation- Intensive
Arithmetic Functions(CIAF) currently implemented in many
Digital Signal Processing (DSP) applications such as
convolution, Fast Fourier Transform(FFT), filtering and in
microprocessors in its arithmetic and logic unit. Since
multiplication dominates the execution time of most DSP
algorithms, so there is a need of high speed multiplier.
Currently, multiplication time is still the dominant factor in
determining the instruction cycle time of a DSP chip. The
demand for high speed processing has been increasing as a
result of expanding computer and signal processing
applications. Higher throughput arithmetic operations are
important to achieve the desired performance in many real-
time signal and image processing applications. One of the key
arithmetic operations in such applications is multiplication and
the development of fast multiplier circuit has been a subject of
interest over decades. Reducing the time delay and power
consumption are very essential requirements for many
applications.
Vedic mathematics covers explanation of several modern
mathematical terms including arithmetic, geometry (plane, co-
ordinate), trigonometry, quadratic equations, factorization and
even calculus.
Vedic Mathematics is the name given to the ancient
system of Indian Mathematics which was rediscovered from
the Vedas between 1911 and 1918 by Sri Bharati
KrsnaTirthaji (1884-1960). According to his research all of
mathematics is based on sixteen Sutras, or word-formulae. For
example, 'Vertically and crosswise` is one of these Sutras.
The objectives of this work are listed below:

i. Design a method to put into effect a high speed Vedic
multiplier using barrel shifter.
ii. Develop an algorithm to implement sutra by modified
design of “Nikhilam Sutra”due to its feature of
reducing the number of partial products.
iii. The barrel shifter used at different levels of design,
drastically reduces the delay when compared to
conventional multipliers.
iv. The hardware implementation of Vedic multiplier
using barrel shifter contributes to adequate
improvement of the speed in order to achieve high
outturn.
II. LITERATURE SURVEY
Multiplication is one of the basic arithmetic operations and it
requires substantially more hardware resources and processing
time than addition and subtraction. In fact, 8.72% of all the
instruction in typical processing units is multiplication.
Comparative study of different multipliers is done for low
power requirement and high speed. “UrdhvaTiryakbhyam”
algorithm of Ancient Indian Vedic Mathematics which is
utilized for multiplication to improve the speed, area
parameters of multipliers. Vedic Mathematics suggests one
more formula for multiplication of large number i.e.
“Nikhilam Sutra” which can increase the speed of multiplier
by reducing the number of iterations .
Vedic mathematic is an ancient technique with
unique approach and it has got different sutras. Here,
“Nikhilam Navatascaramam Dasatah”Sutra is used , which is
efficient in speed of the multiplier. The implementation of an
8-bit Vedic multiplier enhanced in terms of propagation delay
when compared with conventional multipliers. In our design
we have utilized 8-bit barrel shifter which requires only one
clock cycle for “n” number of shifts.
Vedic mathematics is an ancient technique which was used in
the time of Vedas. It has got as many as 16 Sutras that can be
used for different Arithmetic calculation. Vedic Sutras apply
to and cover almost every branch of Mathematics. They apply
even to complex problems involving a large number of
mathematical operations. Vedic mathematics has proved to be
the most robust technique for arithmetic operations. In
contrast, conventional techniques for multiplication provide
significant amount of delay in hardware implementation of n-
bit multiplier. Moreover, the combinational delay of the
design degrades the performance of the multiplier. Application
of the Sutras saves a lot of time and effort in solving the
problems, compared to the formal methods presently in vogue.
Though the solutions appear like magic, the application of the
Sutras is perfectly logical and rational
III. PROPOSED WORK
The high speed implementation of such a multiplier
has wide range of applications in image processing, arithmetic
logic unit and VLSI signal processing. The propagation delay
of array multiplier and conventional Vedic multiplier
implementation on FPGA is very high. Since
propogation delay is high it reduces the speed of the device. In
our design we reduce the propagation delay by implementing
the vedic multiplier on FPGA using barrel shifter.
We assume that the multiplier is ‘X’ and multiplicand
is ‘Y’. Though the designation of the numbers is different but
the architecture implemented is same to some extent for
evaluating both the numbers. Mathematical we solve using
nikhilam sutra.
The hardware deployment is partitioned into three blocks.
i. Base Selection Module
ii. Power index Determinant Module
iii. Multiplier.
The base selection module (BSM) is used to select
the maximum base with respect to the input numbers. The
second sub-module power index determinant(PID) is used to
extract the power index of k1 and k2. The multiplier
comprises of base selection module (BSM), power index
determinant (PID), subtractor, barrel shifter, adder/subtractor
as sub-modules in the architecture.
IV. VEDIC MATHEMATICS
History of Vedic Mathematics:Vedic mathematics is part of
four Vedas (books of wisdom). It is part of Sthapatya- Veda
(book on civil engineering and architecture), which is an upa-
veda (supplement) of Atharva Veda. It covers explanation of
several modern mathematical terms including arithmetic,
geometry (plane, co-ordinate), trigonometry, quadratic
equations, factorization and even calculus. His Holiness
Jagadguru Shankaracharya Bharati Krishna Teerthaji
Maharaja (1884-1960) comprised all this work together and
gave its mathematical explanation while discussing it for
various applications.
Algorithms of Vedic Mathematics: Arithmetic operations
such as addition, subtraction and multiplication are deployed

in various digital circuits to speed up the process of
computation. Arithmetic logic unit is also implemented in
various processor architectures like RISC, CISC etc.,Vedic
mathematics has proved to be the most robust technique for
arithmetic operations. In contrast, conventional techniques for
multiplication provide significant amount of delay in hardware
implementation of n-bit multiplier. Vedic Sutras apply to and
cover almost every branch of Mathematics.
Vedic Multiplication: The proposed Vedic multiplier is based
on the Vedic multiplication formulae (Sutras). These Sutras
have been traditionally used for the multiplication of two
numbers in the decimal number system. The multiplier is
based on an algorithm UrdhvaTiryakbhyam (Vertical &
Crosswise) of ancient Indian Vedic Mathematics.
UrdhvaTiryakbhyam Sutra is a general multiplication formula
applicable to all cases of multiplication. It literally means
“Vertically and crosswise”. It is based on a novel concept
through which the generation of all partial products can be
done with the concurrent addition of these partial products.
V. METHODOLOGY
Assume that the multiplier is ‘X’ and multiplicand is
‘Y’. Though the designation of the numbers is different but the
architecture implemented is same to some extent for
evaluating both the numbers.
The mathematical expression for modified nikhilam sutra is
given below.
P=X*Y= 2k2
* (X+Z2*2k1-k2
) + Z1*Z2…………....(1)
Where k1, k2 are the maximum power index of input numbers
X and Y respectively. Z1 and Z2 are the residues in the
numbers X and Y respectively.
 Base Selection Module
The base selection module has power index
determinant (PID) as the sub-module along with barrel shifter,
adder, average determinant, comparator and multiplexer.
Operation: An input 8-bit number is fed to power index
determinant (PID) to interpret maximum power of number
which is fed to barrel shifter and adder. The output of the
barrel shifter is ‘n’ number of shifts with respect to the adder
output and the input based to the shifter. Now, the outputs of
the barrel shifter are given to the multiplexer with comparator
input as a selection line. The outputs of the average
determinant and the barrel shifter are fed to the comparator.
The required base is obtained in accordance with the
multiplexer inputs and its corresponding selection line.
Figure 1: Base Selection Module
 Power Index Determinant
The input number is fed to the shifter which will shift
the input bits by one clock cycle. The shifter pin is assigned to
shifter to check whether the number is to be shifted or not. In
this power index determinant (PID) the sequential searching
has been employed to search for first ‘1’ in the input number
Figure 2: Power Index Determinant

starting from MSB. If the search bit is ‘0’ then the counter
value will decrement up to the detection of input search bit is
‘1’. Now the output of the decrementer is the required power
index of the input number.
 Multiplier Architecture
The base selection module and the power index
determinant form integral part of multiplier architecture. The
architecture computes the mathematical expression in
equation1.Barrel shifter used in this architecture.
The two input numbers are fed to the base selection
module from which the base is obtained. The outputs of base
selection module (BSM) and the input numbers ‘X’ and ‘Y’
are fed to the subtractors. The subtractor blocks are required to
extract the residual parts z1 and z2. The inputs to the power
index determinant are from base selection module of
respective input numbers. The sub-section of power index
determinant (PID) is used to extract the power of the base and
followed by subtractor to calculate the value. The outputs of
subtractor are fed to the multiplier that feeds the input to the
second adder or subtractor. Likewise the outputs of power
index determinant are fed to the third subtractor that feeds the
input to the barrel shifter. The input number ‘X’ and the
output of barrel shifter are rendered to first adder/subtractor
and the output of it is applied to the second barrel shifter
which will provide the intermediate value. The last sub-section
of this multiplier architecture is the second adder/subtractor
which will provide the required result.

Figure 3: Multiplier Architecture
VI. SIMULATION RESULTS AND DISCUSSION
Here we deals with results obtained by simulation,
synthesis and implementation of Vedic Multiplier module
and their sub modules. Simulation is done in Xilinx ISim
simulator. Synthesis results are obtained using ISE tool.
Implementation is done on Xilinx Spartan-6 CSG324
evolution kit and using ChipScope Pro.
Simulation results of Vedic Multiplier Top module, are
discussed below. Simulation Results of Vedic Multiplier is
shown Figure 4. Inputs “a and b” are forced to module each
of 8-bit. The output “op” is obtained which is of 16-bit after
simulation. The inputs values are forced to the module and
simulation is performed to obtain the output. In the above
figure the simulation result of input “a=125” and “b=75” is
shown. After simulation output “op=9375” is obtained. For
different values of inputs, output is obtained after
simulation.
Figure 4: Simulation results of Vedic Multiplier module
Synthesis Report of Vedic Multiplier module are
synthesized using Xilinx XST synthesis tool for Xilinx
Spartan-6 CSG324 device. The RTL schematics of module
shown in Figure 5
Figure 5: RTL Schematic of Vedic Multiplier
The total power utilization is shown in Figure 6.
After simulation and implementation, the power report is
obtained from Xilinx XPower Analyzer. The total power
utilization of the module obtained after simulation is
0.032W.

Figure 6: Power Analysis Report
Figure 7: ChipScope Pro Result

The ChipScope Pro tool also interfaces with your
Agilent Technologies bench test equipment through the
ATC2 software core. This core synchronizes the ChipScope
Pro tool to Agilent’s FPGA Dynamic Probe add-on option.
This unique partnership between Xilinx and Agilent gives
you deeper trace memory, faster clock speeds, more trigger
options, and system-level measurement capability all while
using fewer pins on the FPGA device.
The ChipScope Pro Result of the Vedic Multiplier Module
is shown Figure 7.
VII. CONCLUSIONS
We achieve a high percentage of reduction in the
propagation delay when compared to array multiplier and
conventional Vedic multiplier implementation on FPGA.
The wide ranges of applications of multiplier unit can be
witnessed in VLSI and signal processing applications. The
project can be extended to the power analysis of the
multiplier.
We conclude that the proposed technique of
multiplication using Urdhva Tiragbyam algorithm and
Nikhilam Sutra algorithm causes less latency when
compared to available techniques in literature. The proposed
technique when implemented for 8x8 bit multiplication, the
delay is found to be reduced on SPARTAN 6. The adoption
of this algorithm for higher bit size multipliers will further
show improvement in speed. Further, higher speeds can be
achieved by making use of pipelining and parallel
processing techniques. This work will increase awareness of
Vedic mathematics techniques in the field of engineering
and delivers high performance in DSP Processors.
The proposed Vedic multiplier architecture exhibits
speed improvements. The 8x8 Vedic multiplier employing
Nikhilam Sutra found to be better than 8x8 conventional
multiplier in terms of speed when magnitude of both
operands are more than half of their maximum values. This
approach may be well suited for multiplication of numbers
with more than 16bit size.
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FPGA Implementation of High Speed 8bit Vedic Multiplier using Barrel Shifter

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FPGA Implementation of High Speed 8bit Vedic Multiplier using Barrel Shifter