Iaetsd low power high speed vedic multiplier using reversible

LOW POWER HIGH SPEED VEDIC MULTIPLIER USING REVERSIBLE
LOGIC GATES: A REVIEW
1
M.Anitha 2
A.Rajani 3
N.Pushpalatha
M.Tech (DECS) Student, Assistant professor, Assistant professor
Department of ECE, AITS
Annamacharya Institute of Technology and Sciences, Tirupati, India-517520
1
anithareddy.me@gmail.com
2
rajanirevanth446@gmail.com
3
pushpalatha_nainaru@rediffmail.com
Abstract-- Multipliers design is difficulty and very
important in DSP (Digital Signal Processing) systems. It
requires more hardware resources and processing time.
Multipliers can perform multiplication and addition
operations. Real-time signal processing requires high
speed and high throughput. Vedic mathematics is one of
the algorithms, it having different sutras to perform
multiplication in easy way. Urdhva Tiryagbhyam Vedic
multiplier is one multiplier that consumes low power,
high speed by using reversible logic, which is always a
key to achieve a high performance digital signal
processing system. In this paper we have to increase the
performance of the previous design. Total reversible
logic implementation cost is reduced by using proposed
method. Multipliers are very important in FFT (Fast
Fourier Transforms), Nano Technology, Embedded
Systems. By using reversible logic number of garbage
outputs, constant inputs, quantum computing is
reduced. In order to achieve low power designs
Quantum computing and reversible circuits are used.
Urdhva Tiryagbhyam Vedic multiplier implementation
using reversible logic can be done by Xilinx software
using Verilog HDL language.
Keywords:- Reversible Logic, Urdhva Tiryagbhyam,
TRLIC (Total Reversible Logic Implementation Cost).
I.INTRODUCTION
The Sanskrit word Veda is derived from the
root Vid, meaning to know without limit. Vedic
mathematics is one of most ancient method to
perform mathematical calculations in simple way. It
can perform large arithmetic operations to simple
mind calculations. The Vedic mathematics having 16
different sutras introduced by Jagadguru swami Sri
Bhakti Krishna Tirtha Maharaja, from that Urdhva
Tiryagbhyam Vedic multiplier is one type of
multiplier which performs crosswise and vertical
operations between the two numbers. This Sutra was
traditionally used in ancient India for the
multiplication of two decimal numbers in relatively
less time and useful in digital hardware.
With the advancement in the VLSI
Technology, there is an ever increasing demand for
the multiplication. Multiplication in DSP is
omnipresent in almost every engineering discipline.
In DSP Faster additions and multiplication operations
are very important and Multiplication is the most
basic and frequently used operations in a CPU
(Central Processing Unit). Multiplication is an
operation performs multiplication between two
binary numbers. Multiplication operations are also
important for other complex operations such as
convolution, Discrete Fourier Transform (DFT), Fast
Fourier Transforms (FFT), etc. DSP engineers are
awaited for new method and hardware
implementation for the multiplication. For the
multiplication DSP engineer has to concentrate on
power dissipation and speed. There is a relation
between power and speed. The reversible logic
computation which takes zero power dissipation. So
reversible logic delay is reduced .The reversible
Urdhva Tiryagbhyam multiplier has been proposed.
The paper is organized as follows: The section II
gives the information about reversible logic along
with the conventional combinational logic. Section
III explains the Urdhva Tiryagbhyam Vedic
multiplier algorithm. The section IV describes the
Urdhva Tiryagbhyam multiplier using conventional
logic. Section V compares the proposed design and
draws a conclusion claiming the versatility of
reversible Urdhva Tiryagbhyam multiplier.
II. REVERSIBLE LOGIC
A. REVERSIBLE LOGIC CIRCUITS
Irreversible circuits (conventional logic
circuits) loosing one bit of information and generates
heat by the second law of thermo dynamics. The
irreversible logic information dissipates (KTln2)
joules of heat energy, where K is Boltzmann's
constant and T is the absolute temperature. The
reversible logic circuits do not dissipate energy as
much as irreversible circuits. The irreversible circuits
mapping between input and output is many- to-one
(number of inputs=single output).Thus, energy
dissipation and the number of bits lost during
computation are in direct prapotion. Threshold
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ISBN:378-26-138420-0234

voltage and power supply applied for the circuit at
that time energy consumption is reduced. In 1973,
Bennett, proved that in order to avoid kTln2 joules of
energy dissipation for a circuit it must be built from
reversible circuits.
In Reversible logic circuits the mapping
between input and output is one-to-one mapping
(number of inputs=number of outputs). In reversible
circuits the input data can uniquely recovered from
output data .By using of this technique there is no
information lost. In order to achieve low power
designs Quantum computing and reversible circuits
are used.
B. REVERSIBLE LOGIC GATES:
The reversible logic gates are n-input and n-
output logic gates. Reversible logic is similar to the
convention logic but reversible logic circuits are
constructing by reversible logic gates. We have
different reversible logic gates are present for the
Urdhva Tiryagbhyam Vedic multiplier.
NOT GATE: 1X1 NOT Gate performs complement
output of present input as shown in fig(1) and gives
zero quantum cost.
Fig(1): Reversible NOT gate
FEYNMAN GATE: 2X2 Feynman Gate mainly used
for fan-out purposes. It performs complementing and
XOR operations are shown in fig(2) . It is also called
as CNOT gate. The quantum cost of Feynman gate is
one.
Fig(2) :Reversible Feynman gate
PERES GATE: 3X3 Peres gate performs AND,
XOR, complementing operations are shown in fig
(3). The quantum cost of Peres gate is four.
Fig(3):Reversible Peres gate
NFT GATE: 3X3 NFT gate performs AND, XOR,
complementing operation and its quantum
implementation is as shown in the fig(4). The
quantum cost of NFT gate is five.
Fig(4):Reversible NFT gate
HNG GATE: 4X4 HNG GATE Performs full adder
operation as shown in fig(5) and provide minimum
quantum cost. The quantum cost is six.
Fig(5):Reversible HNG gate
BVPPG GATE: 5X5 BVPPG Gate performs AND,
XOR, complementing operation and its quantum
implementation is as shown in the fig(6). The
quantum cost of BVPPG gate is ten.
Fig(6):Reversible BVPPG gate
Important parameters of reversible logic:
The most important parameters of the reversible logic
1. Minimized Constant inputs (CI): By using
reversible logic the number of inputs that are to be
maintained constant at either 0 or 1 in order to
synthesize the given logical function.
2. Minimize the garbage: This garbage referred as
number of outputs which are not used in the synthesis
of a required function are very essential, without
reversibility we can’t achieved.
3. Minimize the width of the circuit: In reversible
logic one output given one time as the input of
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ISBN:378-26-138420-0235

another gate .so the width of the circuit is reduced by
number of additional inputs.
4. Minimize the total number of gates: Based on
number of gates usage the delay of the circuit is
reduced. Here we are using less number of gates for
circuit implementation.
5. Quantum cost (QC): Quantum cost depends on the
cost of the circuit in terms of the cost of a primitive
gate. It is calculated knowing the number of primitive
reversible logic gates (1X1 or 2X2) required to
realize the circuit.
6. Total Reversible Logic Implementation Cost
(TRLIC): In a reversible logic circuit TRLIC
calculated as number of garbage outputs, quantum
cost, constant inputs and number of Reversible logic
gates .
TRLIC=∑ (NG+CI+GO+QC)…. (1)
III. URDHVA TIRYAGBHYAM
MULTIPLICATION ALGORITHM
Urdhva Tiryagbhyam multiplier is one type
of multiplier based on Vedic mathematical algorithm.
Urdhva Tiryagbhyam sutra is the General Formula
applicable to all cases of multiplication like binary,
hexa and decimals.The formula itself is very short
and terse, consisting of only one compound word and
means “vertically and cross-wise.” The applications
of this brief and terse sutra are manifold. In this
concept that generation of all partial products and
their additions are performed. This algorithm
performs nXn bit number. The partial products and
their sums are calculated in parallel, the multiplier is
independent of the clock frequency of the processor.
Due to multipliers regular structure, it can be easily
layout in a silicon chip. The Vedic Multiplier based
on this sutra has the advantage that gate delay, the
number of bits increases, and area increases very
slowly as compared to other conventional multipliers.
The binary multiplication of UT algorithm as shown
in fig(7).
Multiply 101 with 110:
1. First take the right hand side digits of 101 and 110
i.e.’1’and ‘0’ multiply both and result will be at
LSB side.
2.Multiply 2nd
right digit of top number with 1st
right digit of bottom number and 1st
right of top
number with 2nd
right of bottom number , add each
other and placed at 2nd
LSB position.
3. Multiply 1st
left of top number with 1st
right of
bottom number, 1st right of top number with 3rd
right
of bottom number and 2nd
digits of both numbers, add
each other and result place at 3rd
LSB position.
4. This step is similar to 2nd
step, move one place to
left and multiply the numbers and place the
result at 4th
LSB position.
5. Finally multiply 1st
right digits of top and bottom
numbers, place that result on next LSB position and
if it having any carry place at MSB position either ‘1’
or ‘0’. That result will be the final product of 101
and 110.
Fig(7): Urdhva Tiryagbhyam procedure for
multiplication algorithm:
IV. EXISTING METHODS
Conventional logic design implementation
of 2X2 Urdhva Tiryagbhyam multiplier with
irreversible logic as show in fig (8). Here we have
four expressions by using irreversible logic and
numbers of gates are seven. The 2X2 multiplier
circuit using reversible logic implementation as
shown in fig (9). The number the circuit uses five
PERES gates and one FEYNMAN (CNOT) gate.
This design has a total quantum cost of 21, number of
Garbage outputs as 11 and number of constant inputs
4. The gate count is 6. This reversible logic does n’t
allow the fan outs. The performance of the Urdhva
Tiryagbhyam multiplier (UT) multiplier is
optimizing each individual unit in terms of quantum
cost, garbage outputs etc.
Fig(8): Conventional 2X2 Urdhva Tiryagbhyam
Multiplier.
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ISBN:378-26-138420-0236

A.2X2 URDHVA TIRYAKBHYAM MULTIPLIER:
The design expressions of multiplier can be
logically modified, so as to optimize the design. This
optimized design makes use of on BVPPG, three
Peres gates and a single FEYNMAN gate. The design
implementation can be optimized using reversible
logic, so the constraints are fan-outs. Other being
loops not permitted. This means that the reversible
logic circuit with multiple numbers of same inputs is
not allowed. In this optimized design One way out is
to use a separate fan out generator or to build a circuit
that inherently takes care of fan outs using the
reversible logic gates used. This optimized design has
a quantum cost of 23, number of garbage outputs as
5, number of gates 5 and the number of constant
inputs is 5.
Fig(9): Design of 2X2 UT multiplier using
reversible logic
Fig(10): Improved Design of 2X2 UT multiplier
using reversible logic -1
The next optimized design also considers the
fan out using BVPPG, two Peres gates, one
Feynman gate and one NFT gate as shown in the
figure 5. The new optimized circuit quantum cost of
the circuit is 24; number of garbage outputs as 4,
number of gates 5 and the number of constant inputs
is 5. I1, I2, I3 (Fig 9&10) and I4 (Fig 10) are the
intermediate outputs that are used for fan-out
purposes
Fig(11): Improved Design of 2X2 UT multiplier
using reversible logic -2
B.4X4 URDHVA TIRYAGBHYAM MULTIPLIER
The reversible 4X4 Urdhva Tiryagbhyam
Vedic multiplier design can be implemented by using
2X2 multiplier. By using four 2X2 multipliers the
4x4 multipliers are implemented as show in fig (11).
Each 2X2 multiplier of which procures four bits as
inputs; two bits from the multiplicand and two bits
from the multiplier. The lower two bits of the output
of the first 2X2 multiplier are entrapped as the lowest
two bits of the final result of multiplication and two
zeros are concatenated with the upper two bits and
given as input to the four bit ripple carry adder. The
other four input bits for the ripple carry adder are
obtained from the second 2X2 multiplier. Likewise
the outputs of the third and the terminal 2X2
multipliers are given as inputs to the second four bit
ripple carry adder. The outputs of these two four bit
ripple carry adders are in turn 5 bits each which need
to be summed up. This five bit ripple carry adder
generates a six bit output. These six bits from the
upper bits of the final result.
C. DESIGN OF RIPPLE CARRY ADDERS
The design of 5 bit ripple carry adder
consists of only HNG gates. By using all HNG gates
quantum cost and garbage outputs are more .so the
ripple carry adder is modified as one Peres gate can
efficiently replace a HNG. Then the number of HNG
gates is 4 and one Peres gate. This 5 bit ripple carry
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ISBN:378-26-138420-0237

adder is used in the second stage of the 4X4 Urdhva
Tiryagbhyam Multiplier. Since for any ripple carry
adder the input carry for the first full adder is zero,
this implicitly means the first adder is a half adder.
Thus a Peres gate HNG. This cut down the quantum
cost by two for any ripple carry adder and the
garbage output by one. The Constant inputs and the
gate count remain constant. The 4X4 UT multiplier
structure is as shown in figure (12).
Fig(11):5-Bit ipple carry adder
Fig(12):4-Bit ripple carry adder
V. RESULTS AND COMPARISONS
The design of the Urdhva Tiryagbhyam
multiplier reversible 2x2 and 4x4 multipliers is
logically verified using XILINX 9.2i and
MODELSIM. The simulation results are as shown in
figures 10 and 11 respectively. The following are the
important design constraints for any reversible logic
circuits.
1. Quantum cost of Reversible logic circuit should be
minimum.
2. Number of garbage outputs of Reversible logic
circuit should be minimum.
3. Number of constant inputs of Reversible logic
4. Number of reversible gates of Reversible logic
Total reversible logic implementation cost calculated
as summing of all constraints and based on above
constraints the total reversible logic implementation
cost is reduced. The 4X4 Vedic multiplier using
reversible logic compared with another multipliers as
shown in table 1. The 4x4 multipliers are take care of
fan outs also. So the quantum cost of 2x2 UT
multipliers quantum cost is increased compare to
previous designs.
Fig(14):Block diagram of 4X4 UT multiplier
Fig(15):Output waveform for 2X2 Multiplier using
Conventional logic .
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ISBN:378-26-138420-0238

CONCLUSIONS
The main aim of UT algorithm is mainly to
design a low power and high speed multipliers using
reversible logic gates. This is the optimized design as
compared to conventional multiplier. The efficiency
of reversible logic circuit is realized in terms of
number of gates, quantum cost, constant inputs, and
garbage outputs. If these parameters are less the
circuit is efficient. By reducing these parameters the
TRLIC is reduced and also lower TRLIC implicitly
means lower the quantum cost, hence lower the
delay and vice versa. Besides combining the design
criterion that fan-out must be generated with the
reversible logic. The further optimization of the
circuit in terms of high speed and low power as
future work.
Table (1): Comparisons of Existing, implemented methods of 2x2 Multiplier and 4x4 Multiplier
ACKNOWLEDGMENTS
We would like to thank our Parents and Teachers
who have been constantly supporting all our works.
REFERENCES
[1] Swami Bharati Krsna Tirtha, Vedic Mathematics.
Delhi: Motilal Banarsidass publishers 1965.
[2]VedicMathematics:
http://www.hinduism.co.za/vedic.html.
[3] R. Landauer,"Irreversibility and Heat Generation
in the Computational Process", IBM Journal of
Research and Development, 5, pp.183-191, 1961.
[4] C.H. Bennett, "Logical reversibility of
Computation", IBM J. Research and Development,
pp.525-532, November
[8] E. Fredkin and T. Toffoli,"Conservative Logic",
Int'l 1 Theoretical Physics Vo121, pp.219-253, 1982.
[9]. Reversible logic gates
http://www.reversible logic gates.com
[10].http://multipliers using reversible logic and
implementation.
[11] Rakshith Saligram and Rakshith T.R. "Novel
Code Converter Employing Reversible Logic",
International Journal of Computer Applications
(IJCA),August2012.
[12] M. Haghparast et al. , "Design of a Novel
Reversible Multiplier Circuit using HNG Gate in
Nanotechnology," in World Applied Science Journal,
Vol. 3, No. 6, pp. 974-978, 2008.
[5] Thapliyal, H., M.B. Srinivas and H.R. Arabnia,
2005, A Reversible Version of 4x4 Bit Array
Multiplier with Minimum Gates and Garbage
Outputs, Int. Conf. Embedded System, Applications
(ESA'05), Las Vegas, USA, pp: 106 114.
[6] M. S. Islam et al. , "Realization of Reversible
Multiplier Circuit," in Information Tech. 1, Vol. 8,
No. 2, pp. 117-121,
2005.
[7] H. R. Bhagyalakshmi, M. K. Venkatesha, “An
Improved Design of a Multiplier using Reversible
Logic Gates,” IJEST, Vol. 2, No. 8, 2010.
[13] G Ganesh Kumarand V Charishma, Design of
high speed vedic multiplier using vedic mathematics
techniques, ltn'l J. of Scientific and Research
Publications, Vol. 2 Issue 3 March 2012
Multiplier design Number of
gates
Constant inputs Garbage
outputs
Quantum cost TRLIC
4X4 UT multiplier
using reversible logic 34 2 Reduced Reduced reduced
2X2 UT multiplier
using Conventional
logic
11 4 11 21 47
Improved Design-1 of
2X2 UT multiplier 5 5 5 23 38
Improved Design-2 of
2X2 UT multiplier 5 5 4 24 38
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ISBN:378-26-138420-0239

Iaetsd low power high speed vedic multiplier using reversible

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Iaetsd low power high speed vedic multiplier using reversible