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International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1537
REALIZATION OF DECIMAL MULTIPLICATION USINGRADIX-16 MODIFIED
BOOTH ENCODINGALGORITHM
N. JAGADEESH CHANDRA MURTHY, E.V. NARYANA
VLSI AND EMBEDDED SYSTEMS, JNTUK
ASST. PROFESSOR, JNTUK
---------------------------------------------------------------------------***---------------------------------------------------------------------------
Abstract - This paper presents the concept of decimal
multiplication using modified booth algorithm. It is
one of the most important decimal arithmetic
operations which have a growing demand in the area
of commercial, financial, and scientific computing. In
this paper, Multiplier is one of the key hardware
blocks in most digital and high performance systems
such as FIR filters, digital signal processors and
microprocessors etc. With advances in technology,
many researchers have tried and are trying to design
multipliers which offer either of the following- high
speed, low power consumption, regularity of layout
and hence less area or even combination of them in
multiplier. Thus making them suitable for various high
speed, low power, and compact VLSI implementations.
However area and speed are two conflicting
constraints. So improving speed results always in
larger areas. So here we try to find out the best trade
off solution among the both of them. Generally as we
know multiplication goes in two basic steps. Partial
product and then addition. Then we turned to Booths
Multiplier and designed Radix-16 modified booth
multiplier and analyzed the performance of
multiplier.
Index Terms— Radix-16 multiplier, Modified recoder(MBR),
Booth encoder(BE),VLSI design.
I. Introduction
Multiplication (Series of repeated additions) is an essential
function in basic arithmetic operations, especially in signal
processing applications, includes graphics and
computation system (Mead, C. and Conway, L., 1988). The
factorization of a large number checks whether a number
is prime. It depends on multiplication. Multiplicand is the
number that is added and the result of it is the product.
The modified stand-alone multiplier consists of a modified
recorder (MBR). MBR has two parts, i.e., Booth Encoder
(BE) and Booth Selector (BS). The operation of BE is to
decode the multiplier signal, and the output is used by BS
to produce the partial product.
Fig1: booth algorithm
II. Modified Booth Algorithm
The multiplier architecture consists of two architectures,
i.e., Modified Booth. By the study of different multiplier
architectures, we find that Modified Booth increases the
speed because it reduces the partial products by half. Also,
the delay in the multiplier can be reduced by using Wallace
tree. The energy consumption of the Wallace Tree
multiplier is also lower than the Booth and the array. The
characteristics of the two multipliers can be combined to
produce a high-speed and low-power multiplier.
The modified stand-alone multiplier consists of a modified
recorder (MBR). MBR has two parts, i.e., Booth Encoder
(BE) and Booth Selector (BS). The operation of BE is to
decode the multiplier signal, and the output is used by BS
to produce the partial product. Then, the partial products
are added to the Wallace tree adders, similar to the carry-
save-adder approach. The last transfer and sum output
line are added by a carry look- ahead adder, the carry
being stretched to the left by positioning.
Booth multiplication algorithm consists of three major
steps as shown in the structure of booth algorithm figure
that includes generation of partial product called as
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1538
recoding, reducing the partial product in two rows, and
addition that gives final product. For a better
understanding of modified booth algorithm & for
multiplication, we must know about each block of booth
algorithm for multiplication process.
III. Modified Booth Algorithm Encoder
This modified booth multiplier is used to perform high-
speed multiplications using modified booth algorithm. This
modified booth multiplier’s computation time and the
logarithm of the word length of operands are proportional
to each other. We can reduce half the number of partial
product. Radix-4 booth algorithm used here increases the
speed of multiplier and reduces the area of multiplier
circuit. In this algorithm, every second column is taken and
multiplied by 0 or +1 or +2 or -1 or -2 instead of
multiplying with 0 or 1 after shifting and adding of every
column of the booth multiplier. Thus, half of the partial
product can be reduced using this booth algorithm. Based
on the multiplier bits, the process of encoding the
multiplicand is performed by radix-4 booth encoder.
Multiplier: B=−bn−12n−1+∑n−2i=0bi2i
 Introduce new variables: b^i=−bi+bi−1 for
i=0…n−1 (assume b−1=0).
 Compute P=∑n−1i=0(b^i×2i×A)
 Although this looks very similar to the normal
product, note that any time bi=bi−1 there is no
addition to be performed as the partial product
will be zero. In the case of serial addition, these
steps can be skipped, thus saving computation
Figure 2:Radix16 modified booth encoder
To reduce the number of partial products added while
multiplying the multiplicand higher radix Booth Encoding
algorithm is one of the most well known techniques used
.Radix 16 Booth algorithm which scan strings of five bits
with the algorithm given below:
(1) Extend the sign bit 1 position if necessary to ensure
that n is even.
(2) Append a 0 to the right of the LSB of the multiplier.
(3) According to the value of each vector, each Partial
Product will be 0,y,+2y,+3y,+4y,+5y,+6y,+7y,+8y, -8Y, -
7y, - 6y, - 5y, - 4y, -3y, - 2Y, - Y. The multiplication of y
is done by shifting y by one bit to the left. Thus, in any case,
in designing n bit parallel multipliers, only n/4 partial
products are generated
IV. MODIFIED BOOTH PROCEDURE
Multiplication is one of the most important and basic
arithmetic operation that constitute programs. In fact
8.72% of all instructions in typical scientific programs are
based on multiplication operation [2]. Many multipliers
have been proposed in the past with consideration of small
area, low power and high performance. Multiplication is
achieved by the addition of a certain number of partial
products rows. Each partial product row is generated by
multiply the multiplier bit one by one to multiplicand. In a
simple multiplier, the generated partial products rows are
equal to the number of bits in multiplier. For example, in
8×8 bit multiplication, it will produce 8 partial product
rows. It will take more adders and more time. To improve
the performance of the multiplier, Booth multiplier is
mostly used multiplier. The number of partial products
rows that must be added to give the multiplications result
can be reduced by using Booth decoding. In Booth
multiplier, the numbers of reduced partial products rows
are depend on the grouping done at multiplier bits. These
groups of multiplier perform the selected operation on
multiplicand. In booth multiplier grouping is done by 2
bits, 3 bits, 4 bits and so on. Higher order booth decoding
reduces the number of partial product rows by a greater
by decoding larger groups of multiplier bits. This
multiplication process is completed in 3 steps. First step:
multiplier bits are divided in groups then these groups are
fed to decoder at where it will indicate that which
operation is to perform on multiplicand. Second step: here
indicated operation performs on the multiplicand and it
will generate the partial products. Third step: Now
generated partial products are adding with adders.
For requirements of smaller area occupation and faster
operation. This is suitable for 2’s complementary and
signed number multiplication.
• Pad the LSB with one zero.
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1539
• Pad the MSB with 2 zeros if n is even and 1 zero if
n is odd.
• Divide the multiplier into overlapping groups of n-
bits based on radix using
• Determine partial product scale factor from
modified booth encoding table.
• Compute the Multiplicand Multiples.
• Sum the Partial Products obtained
ALGORITHM STEPS
• Append ‘0’ to multiplier
• Multiplier recoding
• Booth encoding
• Partial product generation
V. RESULTS
After design entry and optimal simulation, you run
synthesis. During this step Verilog Language designs
become net list files that are accepted as input to the
implementation step. The counters Synthesis & simulation
results are:
Figure 3: RTL schematic of modified radix 16 booth
multiplier
Figure 4. Technological view of Decimal multiplication
Figure.5: Simulation Result of modified radix 16 booth
multiplier
VI. CONCULSION
In Multipliers we studied different Multipliers starting
from Array Multiplier to Wallace Tree, Booth Multipliers,
both Radix-4 and Radix-16.We found that parallel
multipliers are much better than the serial multipliers due
to less area consumption and hence the less power
consumption. Comparing Radix-4 and Radix-16 booth
multipliers we found that radix-16 consumes less power
than radix-4, because radix-16 uses almost half number of
iterations than radix-4.We saw Wallace tree having nearly
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1540
same delay as of radix-16 multipliers where as consuming
a little more power than the former. After all this then we
tried to improve power efficiency of circuits. Hence we
went for studying different recoding schemes along with
their Partial Product generators and study time and power
required by them in a multiplication process. After
studying them we went to modify one of the recoding
schemes to find a proper combination of recoder and PP
generator such that we will have simplest PP generator as
these take maximum area in a cell area and then take care
of zero handling as it handles most of the switching
activities. Hence we ended up creating a better recoding
scheme.
VII. REFERENCES
1. A. Aswal, M.G. Perumal, and G.N.S. Prasanna, “On
basis financial decimal operations on binary
machines,” IEEE trans. Comput., vol. 61, no. 8, pp.
1084-1096, Aug. 2012.
2. M.F. Cowlishaw, E.M. Schwarz, R.M. Smith, and C.F.
Webb, “A decimal floating-point specification,” in
Proc. 15th IEEE symp. Comput.Arithmatic, June
2001, pp. 147-154.
3. M.F. Cowlishaw, “Decimal floating- point:
Algorithm for computers,” in Proc. 16th IEEE
symp. Comput.Arithmatic, Jully 2003.
4. S. Carlough and E. Schwarz, “Power6 decimal
divide,” in Proc. 18th IEEE Symp.
5. S. Carlough, S. Mueller, A. Collura, and M. Kroener,
“The IBM zEnterprise-196 decimal floating point
accelerator”.
6. L. Dadda, “Multioperand parallel decimal adder: A
mixed binary and BCD approach,” IEEE Trans.
Comput., vol. 56, no.10, pp. 1320-1328, oct. 2007.
7. L. Dadda and A. Nannarelli, “A variant of a Radix-
10 combinational multiplier,” in proc. IEEE Int.
Symp.Circuitssyst, May 2008, pp. 3370-3373.
8. L. Han and S. Ko, “High speed parallel decimal
multiplication with redundant internal
encodings,” IEEE Trans. Comput., vol. 62, no. 5, pp.
956–968, May 2013

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IRJET- Realization of Decimal Multiplication using Radix-16 Modified Booth Encoding Algorithm

  • 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1537 REALIZATION OF DECIMAL MULTIPLICATION USINGRADIX-16 MODIFIED BOOTH ENCODINGALGORITHM N. JAGADEESH CHANDRA MURTHY, E.V. NARYANA VLSI AND EMBEDDED SYSTEMS, JNTUK ASST. PROFESSOR, JNTUK ---------------------------------------------------------------------------***--------------------------------------------------------------------------- Abstract - This paper presents the concept of decimal multiplication using modified booth algorithm. It is one of the most important decimal arithmetic operations which have a growing demand in the area of commercial, financial, and scientific computing. In this paper, Multiplier is one of the key hardware blocks in most digital and high performance systems such as FIR filters, digital signal processors and microprocessors etc. With advances in technology, many researchers have tried and are trying to design multipliers which offer either of the following- high speed, low power consumption, regularity of layout and hence less area or even combination of them in multiplier. Thus making them suitable for various high speed, low power, and compact VLSI implementations. However area and speed are two conflicting constraints. So improving speed results always in larger areas. So here we try to find out the best trade off solution among the both of them. Generally as we know multiplication goes in two basic steps. Partial product and then addition. Then we turned to Booths Multiplier and designed Radix-16 modified booth multiplier and analyzed the performance of multiplier. Index Terms— Radix-16 multiplier, Modified recoder(MBR), Booth encoder(BE),VLSI design. I. Introduction Multiplication (Series of repeated additions) is an essential function in basic arithmetic operations, especially in signal processing applications, includes graphics and computation system (Mead, C. and Conway, L., 1988). The factorization of a large number checks whether a number is prime. It depends on multiplication. Multiplicand is the number that is added and the result of it is the product. The modified stand-alone multiplier consists of a modified recorder (MBR). MBR has two parts, i.e., Booth Encoder (BE) and Booth Selector (BS). The operation of BE is to decode the multiplier signal, and the output is used by BS to produce the partial product. Fig1: booth algorithm II. Modified Booth Algorithm The multiplier architecture consists of two architectures, i.e., Modified Booth. By the study of different multiplier architectures, we find that Modified Booth increases the speed because it reduces the partial products by half. Also, the delay in the multiplier can be reduced by using Wallace tree. The energy consumption of the Wallace Tree multiplier is also lower than the Booth and the array. The characteristics of the two multipliers can be combined to produce a high-speed and low-power multiplier. The modified stand-alone multiplier consists of a modified recorder (MBR). MBR has two parts, i.e., Booth Encoder (BE) and Booth Selector (BS). The operation of BE is to decode the multiplier signal, and the output is used by BS to produce the partial product. Then, the partial products are added to the Wallace tree adders, similar to the carry- save-adder approach. The last transfer and sum output line are added by a carry look- ahead adder, the carry being stretched to the left by positioning. Booth multiplication algorithm consists of three major steps as shown in the structure of booth algorithm figure that includes generation of partial product called as
  • 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1538 recoding, reducing the partial product in two rows, and addition that gives final product. For a better understanding of modified booth algorithm & for multiplication, we must know about each block of booth algorithm for multiplication process. III. Modified Booth Algorithm Encoder This modified booth multiplier is used to perform high- speed multiplications using modified booth algorithm. This modified booth multiplier’s computation time and the logarithm of the word length of operands are proportional to each other. We can reduce half the number of partial product. Radix-4 booth algorithm used here increases the speed of multiplier and reduces the area of multiplier circuit. In this algorithm, every second column is taken and multiplied by 0 or +1 or +2 or -1 or -2 instead of multiplying with 0 or 1 after shifting and adding of every column of the booth multiplier. Thus, half of the partial product can be reduced using this booth algorithm. Based on the multiplier bits, the process of encoding the multiplicand is performed by radix-4 booth encoder. Multiplier: B=−bn−12n−1+∑n−2i=0bi2i  Introduce new variables: b^i=−bi+bi−1 for i=0…n−1 (assume b−1=0).  Compute P=∑n−1i=0(b^i×2i×A)  Although this looks very similar to the normal product, note that any time bi=bi−1 there is no addition to be performed as the partial product will be zero. In the case of serial addition, these steps can be skipped, thus saving computation Figure 2:Radix16 modified booth encoder To reduce the number of partial products added while multiplying the multiplicand higher radix Booth Encoding algorithm is one of the most well known techniques used .Radix 16 Booth algorithm which scan strings of five bits with the algorithm given below: (1) Extend the sign bit 1 position if necessary to ensure that n is even. (2) Append a 0 to the right of the LSB of the multiplier. (3) According to the value of each vector, each Partial Product will be 0,y,+2y,+3y,+4y,+5y,+6y,+7y,+8y, -8Y, - 7y, - 6y, - 5y, - 4y, -3y, - 2Y, - Y. The multiplication of y is done by shifting y by one bit to the left. Thus, in any case, in designing n bit parallel multipliers, only n/4 partial products are generated IV. MODIFIED BOOTH PROCEDURE Multiplication is one of the most important and basic arithmetic operation that constitute programs. In fact 8.72% of all instructions in typical scientific programs are based on multiplication operation [2]. Many multipliers have been proposed in the past with consideration of small area, low power and high performance. Multiplication is achieved by the addition of a certain number of partial products rows. Each partial product row is generated by multiply the multiplier bit one by one to multiplicand. In a simple multiplier, the generated partial products rows are equal to the number of bits in multiplier. For example, in 8×8 bit multiplication, it will produce 8 partial product rows. It will take more adders and more time. To improve the performance of the multiplier, Booth multiplier is mostly used multiplier. The number of partial products rows that must be added to give the multiplications result can be reduced by using Booth decoding. In Booth multiplier, the numbers of reduced partial products rows are depend on the grouping done at multiplier bits. These groups of multiplier perform the selected operation on multiplicand. In booth multiplier grouping is done by 2 bits, 3 bits, 4 bits and so on. Higher order booth decoding reduces the number of partial product rows by a greater by decoding larger groups of multiplier bits. This multiplication process is completed in 3 steps. First step: multiplier bits are divided in groups then these groups are fed to decoder at where it will indicate that which operation is to perform on multiplicand. Second step: here indicated operation performs on the multiplicand and it will generate the partial products. Third step: Now generated partial products are adding with adders. For requirements of smaller area occupation and faster operation. This is suitable for 2’s complementary and signed number multiplication. • Pad the LSB with one zero.
  • 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1539 • Pad the MSB with 2 zeros if n is even and 1 zero if n is odd. • Divide the multiplier into overlapping groups of n- bits based on radix using • Determine partial product scale factor from modified booth encoding table. • Compute the Multiplicand Multiples. • Sum the Partial Products obtained ALGORITHM STEPS • Append ‘0’ to multiplier • Multiplier recoding • Booth encoding • Partial product generation V. RESULTS After design entry and optimal simulation, you run synthesis. During this step Verilog Language designs become net list files that are accepted as input to the implementation step. The counters Synthesis & simulation results are: Figure 3: RTL schematic of modified radix 16 booth multiplier Figure 4. Technological view of Decimal multiplication Figure.5: Simulation Result of modified radix 16 booth multiplier VI. CONCULSION In Multipliers we studied different Multipliers starting from Array Multiplier to Wallace Tree, Booth Multipliers, both Radix-4 and Radix-16.We found that parallel multipliers are much better than the serial multipliers due to less area consumption and hence the less power consumption. Comparing Radix-4 and Radix-16 booth multipliers we found that radix-16 consumes less power than radix-4, because radix-16 uses almost half number of iterations than radix-4.We saw Wallace tree having nearly
  • 4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 09 | Sep 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1540 same delay as of radix-16 multipliers where as consuming a little more power than the former. After all this then we tried to improve power efficiency of circuits. Hence we went for studying different recoding schemes along with their Partial Product generators and study time and power required by them in a multiplication process. After studying them we went to modify one of the recoding schemes to find a proper combination of recoder and PP generator such that we will have simplest PP generator as these take maximum area in a cell area and then take care of zero handling as it handles most of the switching activities. Hence we ended up creating a better recoding scheme. VII. REFERENCES 1. A. Aswal, M.G. Perumal, and G.N.S. Prasanna, “On basis financial decimal operations on binary machines,” IEEE trans. Comput., vol. 61, no. 8, pp. 1084-1096, Aug. 2012. 2. M.F. Cowlishaw, E.M. Schwarz, R.M. Smith, and C.F. Webb, “A decimal floating-point specification,” in Proc. 15th IEEE symp. Comput.Arithmatic, June 2001, pp. 147-154. 3. M.F. Cowlishaw, “Decimal floating- point: Algorithm for computers,” in Proc. 16th IEEE symp. Comput.Arithmatic, Jully 2003. 4. S. Carlough and E. Schwarz, “Power6 decimal divide,” in Proc. 18th IEEE Symp. 5. S. Carlough, S. Mueller, A. Collura, and M. Kroener, “The IBM zEnterprise-196 decimal floating point accelerator”. 6. L. Dadda, “Multioperand parallel decimal adder: A mixed binary and BCD approach,” IEEE Trans. Comput., vol. 56, no.10, pp. 1320-1328, oct. 2007. 7. L. Dadda and A. Nannarelli, “A variant of a Radix- 10 combinational multiplier,” in proc. IEEE Int. Symp.Circuitssyst, May 2008, pp. 3370-3373. 8. L. Han and S. Ko, “High speed parallel decimal multiplication with redundant internal encodings,” IEEE Trans. Comput., vol. 62, no. 5, pp. 956–968, May 2013