Time Efficient Square and Cube Architecture using Vedic Sutras

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Harshdeep Singh Fri, Apr 08, 2022 01:11 PM UTC

This PDF contains :
ABSTRACT,
KEYWORDS,
I.INTRODUCTION,
II. RELATED WORKS,
III.MODIFIED SYSTEM,
IV.OVERVIEW OF THE MODIFIED SYSTEM,
V. ARCHITECTURE OF THE PROPOSED SYSTEM,
VI. RESULTS,
VII.CONCLUSION

1. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
Time Efficient Square and Cube Architecture
using Vedic Sutras
Chinchu R 1 , Nishi.G.Nampoothiri2
PG Scholar, Dept. o f ECE, Musaliar Co llege of Engineering and Technology, Pathanamthitta, Kerala, India1
Associate professor, Dept. of ECE, Musaliar College of Eng ineering and Technology, Pathanamthitta, Kerala, India2
ABSTRACT: Ved ic Mathematics is an ancient system of mathematics. Square and cube are frequently performed
functions in most of the DSP systems. The existing system of the thesis was based on low power square and cube
architecture of 8 bit using Vedic sutras due to its frequent usage. Squaring utilised Duplex property properties of
Urdhva Tiryagbhyam and cubing used Anurupyena sutra. The proposed system aimed at performing both these square
and cube architectures using a single algorith m wh ich is time efficient and ha ve simple architecture based on Vedic
mathematics. The modified system enabled performing 4-bit mu ltip licat ion and addition besides 8-bit squaring and
cubing. Thus the thesis comprises of a single architecture that can perfom addition, multip lication and also special case
of multip licat ion that is square and cube of 8-bit numbers. It basically consists of a control unit and an arithmetic and
logic unit. The control unit selects which operation is to be performed and accordingly the ALU unit functions and
gives the result after the required number of clock cycles. The architecture is simp le and time efficient.
KEYWORDS: Vedic Mathematics, Urdhva Tiryagbhyam, Anurupyena sutra, Duplex property.
I.INTRODUCTION
Vedic mathematics is the name given to ancient mathematical system which was rediscovered fro m the Vedas by Sri
Bharati Krishna Tirthaji between 1911 and 1918. The most important feature of the Vedic mathematics system is its
coherence. Instead of lengthy unrelated techniques the entire system is beautifully interrelated and unified. The general
mu ltip licat ion method can be easily reversed to allow one-line div ision also the simple squaring method can be
reversed to get one-line square root. All these methods can be easily understood. The unifying quality of this system is
its highlight, this makes mathematics easy and it will encourage innovation. In the past, conventional methods have
been used for mult iplication. Conventional methods have been highly time consuming. Heart of many of the DSP
operations like Decoding, Image Co mpression , Demodulation are cube and square architectures. It has numerous
applications in cryptography also. Since many of the DSP sytems use square and cube operations it caused gre ater
delay in the entire system.
The objective of this paper is to design a single architecture for performing square and cube operations due to the wide
usage of these mathemat ical operations in many digital signal processing systems as mentioned above. In the existing
systems two architectures were used for square and cube operations.The main objectives of this thesis are Designing a
single architecture for various mathemat ical operations such as 4-bit mu ltip licat ion and addition and 8-bit square and
cube architecture. Other objectives include: architecture should be simp le as well as time efficient . When opting for the
hardware imp lementation it should be cost effective.The architecture should be consuming less area with respect to the
existing designs.
II. RELATED WORKS
Various methods of operation are used to perform square and cube architecture using Ved ic mathematics in previous
studies included in this field. They include:
1. To calculate the square of a number Duplex property of Urdhva Tiryagbhyam is used. In the Duplex, calculate twice
the product of the outermost pair, then add twice the product of the next outermost pair, and continue till no pairs are
left. If the number of bits is odd one bit will be left in the middle add its squa re[13].
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9202

2. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
2. Square can also be calculated using Duplex property of binary nu mbers wh ich is similar to the Duplex property of
decimal nu mbers present in Vedic mathematics In this Duplex, take twice the product of the outermost pair, then add
twice the product of the next outermost pair, and till no pairs are left. If the number of bits is odd, one bit is left in the
middle, enter it as such.
3. Calculat ion of cube is based on the Anurupya Sutra. It states that begin with the cube of first digit then take ne xt
three numbers in Geo metrical Proportion then the 4th figure present at the right end will be the cube of the second
digit. If a and b are two d igits, according to the Anurupya Sutra the result will be equal to (a+b)³[13].
4. Yavadunam Sutra can also be applied for performing squaring as well as cubing operations of a system.
5. Conventional 8-bit mu ltip liers, 4-bit mu ltip liers etc are replaced by Vedic mu lt ipliers as well.
III.MODIFIED S YSTEM
The work is main ly aimed at 8-b it square and cube architecture. Suitable modifications are done to the existing system
which was based on dwandwa yoga property of urdhvatiryagbhyam sutra for squaring and Yavadunam Sutra for
cubing inorder to confine both the operations in one architecture whereas separate architectures were required for
square and cube in the every existing systems. Also the designing of the architecture was done carefully inorder to
improve the timing efficiency of the system fro m the previous architectures for the square and cube operations. It uses
basic operations like selection, shift ing, addition only. So the arch itecture is simple with mu x, adders, shifters etc.
Along with 8-bit squaring and cubing this design will a lso perform 4-bit addition and multip licat ion. This forms the
modified system.
IV.OVERVIEW OF THE MODIFIED S YS TEM
The modified system is composed of two important blocks. One is the control unit and the other one is the Arithmet ic
and Logic Unit. As the name suggests the ALU performs the arithmetic operations and it is the control unit that
determines which operation is to be performed. The overview of the system is shown in the figure 1.
Figure 1. Block diagram of the proposed system
This proposed design can perform 8-b it square and cube operations. Also 4-bit addition ,4-b it multip licat ion, 4-b it
square and cube operation is possible in this design. Suppose an 8-bit input is given to the system and it is divided into
4-bit MSB,a and 4-bit LSB,b. This two 4-bit numbers can undergo multip lication of the patterns ab, 2ab, 3a²b, 3ab²,
a²b, ab², a²,b²,a³ and b³. Also addition of the 4-bit numbers a and b is also possible. This much variety of operations can
be performed in this design and it is this control unit that determines which operation is to be performed. The control
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9203

3. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
unit is provided with many commands. The command given to the control unit determines the operation to be
performed by the arith metic and logic unit of the proposed design.
TABLE 1 COMMANDS AND OPERATION IN CU
COMMAND OPERATION
8‟h00 (MSB²)(LSB)
8‟h10 (MSB)(LSB)
8‟h20 (MSB³)
8‟h1A (MSB²)
8‟h2A (LSB³)
8‟h3A (LSB²)
8‟h4A (MSB)(LSB²)
8‟h5A (MSB)+(LSB)
8‟hAA INPUT²
8‟hFF INPUT³
ALU as the name suggests is the section which performs all the arith metic operations of the design. It can perform 8-b it
square and 8-bit cube operations from the single architecture which was the main objective of our proposed design.
Besides this it can perform various 4-bit operations shown in the above table apart from the 8-b it operations. It can
perform 4-bit addit ion, mult iplication, squaring and cubing operations. The selection of the operation is based on the
commands given in the table 1.
V. ARCHIT ECTURE OF THE PROPOS ED S YSTEM
The figure 2 shown below is the algorithm showing the operation upto obtaining the partial products
(MSB)(LSB),(MSB)²(LSB), (MSB)(LSB)², (M SB)², (LSB)²,(MSB)³, (LSB)³. The algorith m starts with initially storing
the 8-bit input into a D-flipflop. The 8-b it input is then split into two 4-bit parts ie., 4-bit LSB and 4-bit MSB. This
MSB and LSB are connected to two mult iplexers. This multip lexers according to the select lines S1 and S2 selects
MSB or LSB. If S1 of MUX1 selects MSB then the output of the sq uaring section will be (MSB)² and is S1 of MUX1
selects LSB then its otput will be (LSB)². Similarly according to the select line S2 of M UX2 the output of the
mu ltip lexer will be (M SB) or (LSB). Then the output of the multip lexer with select line S3 of MUX3 will be (M SB)² or
(LSB)² or otherwise (MSB) o r (LSB). The output of MUX2 and MUX3 are given to the vedic mu ltiplier. Also the
output of the MUX3 is also given to MUX6.
The output of the multiplier is stored in a D flip flop so that output doesnot change with the change in the output. The
output of the DFF is given to a leftshifter and an adder. The input given to the left shifter is also given to the MUX6.
The input to the MUX4 is the combination of the output of the left shifter and adder. The output of the MUX4 is
selected by the select line S4. It is given to the MUX 5 wh ich is controlled by the select line S5. Output of the MUX5 is
stored in a 16 b it DFF which forms the partial product of the system. The value stored in the 16 bit DFF may be
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9204

4. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
(MSB)²(LSB), (MSB)(LSB)², (MSB)², (LSB)²,(MSB)³, (LSB)³ or (MSB)(LSB). This value depends on the select line
S1,S2,S3,S4,S5, S6.
Figure 2 A lgorith m showing partial product of the proposed system
The 16-bit partial product stored in the D flipflop as shown in Figure 2 is then coined with 6 zero bits in the MSB to
form a 22 bit value and it is stored in a 22 bit DFF as shown in Figure 3. Consider the 22 bit number is Q[21:0], fro m
this Q[11:0] is stored in DFF1. The output of DFF1 is stored in DFF2 inorder match with the time delay. Q[17:0] is
stored in DFF4 along with [3:0] b its from the output of DFF2. The output of DFF4 is given to a ripple carry adder
RCA2, the inputs to RCA2 are [7:0] bits of the output DFF4 and [21:8] bits of the output DFF4. The output of RCA2
is given as input to DFF3 along with Q[21:18] as LSB. The output of DFF3 is given to RCA 1 by divid ing the input
into [9:0]bits and [21:10] bits.
The output of RCA1 is given to DFF6 and the output of RCA2 is given to DFF7. The input of DFF6 and DFF7 are
given as inputs to the MUX7. The output of MUX7 depends on the select line S. Its output is given to DFF5 which is
given as input to RCA 3 with the bits split into two similar as RCA1. The output of RCA 3, DFF6, and DFF7 are stored
in a D flipflop which is named as CUBEOUT in the figure . The output of RCA3, DFF2 and DFF7 are stored in a D
flipflop named SQUAREOUT. The value stored in the CUBEOUT is the cube output of the given 8-bit input and the
value stored in the SQUA REOUT is the square output of the given 8-bit input. CUBEOUT and SQUA REOUT are
given as input to MUX8 whose select line is s. This select line determines whether to output square or cube depending
on the commands given to the control unit. Also it can perform 4-bit mu ltip licat ion and 4-bit addition operations
depending on the commands of control unit as discussed in the control unit section.
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9205

5. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
Figure 3 A lgorith m showing the operation of the proposed system
Thus, the 8-bit number is given as input as shown in figure 2 and a partial product is obtained as the output of the
algorith m. Then it is given as input of the algorithm shown in figure 3 and we can obtain the final output at the end of
operations as shown in the algorith m.
VI. RES ULTS
The simulation results of the proposed system includes the simu lation of the single architecture showing both the
square and cube output along with multip licat ion and addition operations. The simulat ion result shown in figure 4
includes the square and cube output and also the 4-bit mult iplication output. . In the figure the result named A is the
input given to the architecture, „in‟ represents the commands in the control unit, clk is the clock cycle and Y is the
output of the existing square and cube architecture.
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9206

6. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
Figure 4 Proposed architecture along with 4-bit mu ltip lier
The final simu lation results of the proposed system shown in figure 5 includes the 8-bit square and cube output and also
the 4-bit mult iplication and 4-bit addition output of the system. . In the figure the result named A is the input given to
the architecture, „in‟ represents the commands in the control unit, clk is the clock cycle and Y is the output of the
existing square and cube architecture.
Figure 4 Proposed architecture along with 4-bit mu ltip lier and adder
Fro m the synthesis reports it can also be inferred that d elay occurring in the existing square system is 5.7ns wh ich is
lesser than the conventional methods whose delay is 27ns. Delay occurring in the existing cube system is 7ns which is
lesser than the conventional methods whose delay is 41ns. Delay occurring in the proposed system for the calcu lation
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9207

7. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
of square and cube is 3ns only which has a higher margin than the conventional methods whose delays were 27ns and
41ns respectively.
Figure 5 Time delay co mparison graph of three systems
Synthesis reports shows that for calculating the cube of an 8-bit input the architecture utilizes 349 nu mber of LUTs.
This LUTs are used for the calculation of cube operation alone. For calculating the square of an 8-bit input the
architecture utilizes 58 number of LUTs. This LUTs are used for the calculation of square operation only. Thus for
calculating square and cube simultaneously a total of 407 LUTs are required. For the proposed system,the architecture
utilizes 349 nu mber of LUTs. This LUTs are used for the calculation of square,cube, multiplication and addition
operations simultaneously. So the proposed system utilises less number of LUTs as compared to that of the existing
system. Hence it uses less area than the previous architectures .
Figure 6 Area co mparison graph of three systems
Fro m the graphs it can be inferred that the whole system consumes less area and is time efficient.
Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9208

8. ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical,
Electronics and Instrumentation Engineering
(An ISO 3297: 2007 Certified Organization)
Vol. 4, Issue 11, November 2015
VII.CONCLUS ION
In this work designing and the simu lation of a single architecture for performing 8-bit square and 8-bit cube operations
is main ly done. The system is designed in such a way to include multiplication and addition operation of 4-bit numbers
also. The main purpose of the thesis is the time efficient operation of the design in order to increase the overall
performance of the system utilising this design. Digital signal processing systems are the systems mainly utilising
square and cube operations. It is the bottleneck of many of the DSP operations hence by increasing the speed o f square
and cube operations we can imp rove the overall performance. The aim o f this project was to achieve two objectives.
Firstly,to propose a new architecture to imp rove the timing efficiency of the square and cube operation. Secondly, to
ensure that the both the operations is performed using a single architecture . A lso the architecture designed should be
simp le and utilises less area. In contradiction to the previous systems square and cube operations were done using a
single architecture which is time efficient, simple and consumes less area.
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[12] Swami Bharati Krisna Tirtha, “Vedic Mathematics,” Motilal Banarsidass Publishers, Delhi, 1965.
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Copyright to IJAREEIE DOI:10.15662/IJAREEIE.2015.0411056 9209

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