The booth algorithm is a multiplication algorithm that allows us to multiply the two signed binary integers in 2's complement, respectively. It is also used to speed up the performance of the multiplication process. It is very efficient too. It works on the string bits 0's in the multiplier that requires no additional bit only shift the right-most string bits and a string of 1's in a multiplier bit weight 2^k to weight 2^m that can be considered as 2^{k+ 1} - 2^m.

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Following is the pictorial representation of the Booth's Algorithm:

Booth’s multiplication algorithm is based on the fact that fewer partial products are needed to be generated for consecutive ones and zeros. For consecutive zeros, a multiplier only needs to shift the accumulated result to the right without generating any partial products. Booth algorithm gives a procedure for multiplying binary integers in signed 2’s complement representation in efficient way, i.e., less number of additions/subtractions required. It operates on the fact that strings of 0’s in the multiplier require no addition but just shifting and a string of 1. This video you will learn about Booth's Algorithm Multiplication and Division. This is our Semi Finals/Case Study in Computer Organization (CCS24). Booth’s algorithm is a multiplication algorithm that multiplies two signed binary numbers in 2’s compliment notation. Booth’s algorithm is a multiplication algorithm that multiplies two signed binary numbers in 2’s compliment notation.

In the above flowchart, initially, AC and Q_{n + 1} bits are set to 0, and the SC is a sequence counter that represents the total bits set n, which is equal to the number of bits in the multiplier. There are BR that represent the multiplicand bits, and QR represents the multiplier bits. After that, we encountered two bits of the multiplier as Q_n and Q_{n + 1}, where Qn represents the last bit of QR, and Q_{n + 1} represents the incremented bit of Qn by 1. Suppose two bits of the multiplier is equal to 10; it means that we have to subtract the multiplier from the partial product in the accumulator AC and then perform the arithmetic shift operation (ashr). If the two of the multipliers equal to 01, it means we need to perform the addition of the multiplicand to the partial product in accumulator AC and then perform the arithmetic shift operation (ashr), including Q_{n + 1}. The arithmetic shift operation is used in Booth's algorithm to shift AC and QR bits to the right by one and remains the sign bit in AC unchanged. And the sequence counter is continuously decremented till the computational loop is repeated, equal to the number of bits (n).

Working on the Booth Algorithm

1. Set the Multiplicand and Multiplier binary bits as M and Q, respectively.
2. Initially, we set the AC and Q_{n + 1} registers value to 0.
3. SC represents the number of Multiplier bits (Q), and it is a sequence counter that is continuously decremented till equal to the number of bits (n) or reached to 0.
4. A Qn represents the last bit of the Q, and the Q_n+1 shows the incremented bit of Qn by 1.
5. On each cycle of the booth algorithm, Q_n and Q_{n + 1} bits will be checked on the following parameters as follows:
   1. When two bits Q_n and Q_{n + 1} are 00 or 11, we simply perform the arithmetic shift right operation (ashr) to the partial product AC. And the bits of Qn and Q_{n + 1} is incremented by 1 bit.
   2. If the bits of Q_n and Q_{n + 1} is shows to 01, the multiplicand bits (M) will be added to the AC (Accumulator register). After that, we perform the right shift operation to the AC and QR bits by 1.
   3. If the bits of Q_n and Q_{n + 1} is shows to 10, the multiplicand bits (M) will be subtracted from the AC (Accumulator register). After that, we perform the right shift operation to the AC and QR bits by 1.
6. The operation continuously works till we reached n - 1 bit in the booth algorithm.
7. Results of the Multiplication binary bits will be stored in the AC and QR registers.

There are two methods used in Booth's Algorithm:

1. RSC (Right Shift Circular)

It shifts the right-most bit of the binary number, and then it is added to the beginning of the binary bits.

2. RSA (Right Shift Arithmetic)

It adds the two binary bits and then shift the result to the right by 1-bit position.

Example: 0100 + 0110 => 1010, after adding the binary number shift each bit by 1 to the right and put the first bit of resultant to the beginning of the new bit.

Example: Multiply the two numbers 7 and 3 by using the Booth's multiplication algorithm.

Ans. Here we have two numbers, 7 and 3. First of all, we need to convert 7 and 3 into binary numbers like 7 = (0111) and 3 = (0011). Now set 7 (in binary 0111) as multiplicand (M) and 3 (in binary 0011) as a multiplier (Q). And SC (Sequence Count) represents the number of bits, and here we have 4 bits, so set the SC = 4. Also, it shows the number of iteration cycles of the booth's algorithms and then cycles run SC = SC - 1 time.

The numerical example of the Booth's Multiplication Algorithm is 7 x 3 = 21 and the binary representation of 21 is 10101. Here, we get the resultant in binary 00010101. Now we convert it into decimal, as (000010101)₁₀ = 2*4 + 2*3 + 2*2 + 2*1 + 2*0 => 21.

Example: Multiply the two numbers 23 and -9 by using the Booth's multiplication algorithm.

Here, M = 23 = (010111) and Q = -9 = (110111)

Q_{n + 1} = 1, it means the output is negative.

Hence, 23 * -9 = 2's complement of 111100110001 => (00001100111)

• Open Access
• Total Downloads : 763
• Authors : Snehal R Deshmukh, Dinkar L Bhombe
• Paper ID : IJERTV3IS21104
• Volume & Issue : Volume 03, Issue 02 (February 2014)
• Published (First Online): 03-03-2014
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License

Performance Comparison of Different Multipliers using Booth Algorithm

Snehal R Deshmukh Dept of E&TC SSGMCOE

Booth 27s Algorithm Calculator Instructions

Shegaon, India (MS)

Dinkar L Bhombe Dept of E&TC SSGMCOE

Shegaon, India (MS)

Abstract Low power consumption and smaller area are some of the most important criteria for the fabrication of DSP systems and high performance systems. Optimizing the speed and area of the multiplier is a major design issue. However, area and speed are usually conflicting constraints so that improving speed results mostly in larger areas. In this paper we try to determine the best solution to this problem by comparing a few multipliers. The parallel multipliers like radix 2 and radix 4 modified booth multiplier does the computations using lesser adders and lesser iterative steps. As a result of which they occupy lessr space as compared to the serial multiplier. This a very important criteria because in the fabrication of chips and high performance system requires components which are as small as possible.

Keyswords: multiplier, radix-R, booth algorithm.

1. INTRODUCTION

   Multipliers play an important role in todays digital signal processing and various other applications. With advancements in technology, many researchers have already tried and are still trying to design multipliers which provides either greater speed, less power consumption, regularity of layout and hence small area or even combination of them in one multiplier which makes them suitable for various increased speed, minimized power and compact VLSI implementation. The usual multiplication method is add and shift algorithm. In parallel multipliers number of partial products that needs to be added is the main parameter that defines the performance of the multiplier. In order to minimize the number of partial products to be added, Booth algorithm and Modified Booth algorithm is one of the most popular algorithms [1].

2. MULTIPLIERS

   A Binary multiplier is an electronic hardware circuit that used in digital electronics or a computer or other electronic devices to perform rapid multiplication of two numbers in binary representation. It is obtained using binary adders.

   The rules for binary multiplication are as follows

   1. If the multiplier digit is a 1, then the product will be same as multiplicand and simply it will be copied down.

   2. If the multiplier digit is a 0 the product is also 0.

      The multiplication algorithm for an N bit multiplicand by N bit multiplier is shown in fig1.

      Y= Yn-1 Yn-2……………………Y2 Y1 Y0 Multiplicand

      X= Xn-1 Xn-2 ………………… X2 X1 X0 Multiplier

      Figure 1. Multiplication algorithm for N*N bit

3. TYPES OF MULTIPLIER

   Basically there are three types of multipliers. They are as follows.

   1. Serial Multiplier

      Serial multiplier generates partial products sequentially and adds each newly generated product to previously accumulated partial product.

      Figure 2.Serial Multiplier

      results are then stored in the output register after completing N+M cycles.

      Figure 3

      Serial multiplier is used where area and power is important, &delay can be tolerated. Circuit uses one adder to add the m *n partial products. The circuit is shown for m=n=4. Multiplicand and Multiplier inputs have to be arranged in a special manner synchronized with circuit behavior as shown in the fig 2. The inputs could be presented at different rates depending on the length of the multiplicand and the multiplier. Two clocks are used, one to clock the data & one for the reset. A first order approximation of the delay is 0 (m,n). With this circuit arrangement the delay is given as D=[(m+1)n+1] tfa. As shown in fig 3 the individual PP is formed. The addition of the PPs are performed as the intermediate values of PPs, addition are stored in the D Flip Flop, circulated and then added together with the newly formed PP [2].

      Disadvantage

      This approach is not suitable for larger values of M & N.

   2. Parallel multiplier

      Generates partial products in parallel, accumulates using a fast multi-operand adder.

      Serial/Parallel Multiplier

      Figure 4.Serial/Parallel Multiplier

      One operand is fed to the circuit in parallel while the other is in serial. N partial products are formed for each cycle. On successive cycles, each cycle does the addition of one column of the multiplication table of M*N PPs. The final

      Disadvantage

      Area required is N-1 for M=N.

      Figure 5. Generation of individual PP and their addition

   3. Array Multiplier

      Array of identical cells generating new partial products and accumulating them simultaneously is as shown in figue 6. No separate circuits for generation and accumulation is required. This implementation reduces execution time but increases hardware complexity.

      Figure 6. Array Multiplier

      Array multiplier is well known due to its regular structure. Multiplier is based on add and shift algorithm. Each and every partial product is generated by the multiplication of the multiplicand with one multiplier bit. The partial product are shifted according to their bit orders and then added. The addition can be performed with normal carry propagate adder. N-1 adders are required where N is the multiplier length. Although the method is simple as it can be seen from this example, the addition is done serially as well as in parallel.

      Disadvantage

      Hardware complexity increases with N*M. Now as both multiplicand and multiplier may be positive or negative, 2s complement number system is used to represent them. If the multiplier operand is positive then essentially the same technique can be used but care must be taken for sign bit extension. The reason for dealing with signed number incorrectly is the absence of sign bit expansion in this multiplier.

4. MULTIPLICATION ALGORITHM

   A circuit that multiplies two unsigned n bit binary numbers, uses a 2 dimensional array of identical subcircuits. Each of which contains a full adder and an and gate. For large number of bits this approach may not be appropriate because of the large number of gates needed. Another approach is to use shift register in combination with an adder to implement the traditional method of multiplication. [1]

   P=0;

   For i=0 to n-1 do If bi=1 then

   P=P+A;

   End if;

   Left shift A;

   End for;

   Figure 7. Data circuit of multiplier

5. BOOTH MULTIPLIERS

   This algorithm was invented by Andrew Donald Booth in 1950 while doing study on crystallography. Booth used reception desk calculators that shifts faster than adding and formed the algorithm to increasing the speed. Booth's algorithm is important in the study of computer architecture.[3] . It is a powerful algorithm for signed- number multiplication, which considers both positive and negative numbers uniformly [4]. Booths Algorithm is a smart move for multiplying signed numbers. It starts with the ability to both add and subtract.[5] An algorithm that uses twos complement notation of signed binary numbers for multiplication.[6]

6. MODIFIED BOOTHS ALGORITHM

Modified Booths is two times faster than Booths algorithm. Modified Booth encoding algorithm is an efficient way to reduce the number of partial products by grouping consecutive bits in one of the two operands to form the signed multiples. The operand that is Booth encoded is called the multiplier and the other operand is called the multiplicand. [7]

1. Radix-2

   Booth algoithm gives a procedure for multiplying binary integers in signed 2s complement representation. [3] Illustration of the booth algorithm with example:

   Example, 2 ten x (- 4) ten 0010 two * 1100 two Example, 2 ten x (- 4) ten 0010 two * 1100 two

   Step 1: Making the Booth table [3]

   1. From the above two numbers, pick the number with the smallest difference between a series of consecutive numbers, and make it a multiplier.

   Therefore, multiplication of 2 x ( 4), where 2 ten (0010 two) is the multiplicand and ( 4) ten (1100two) is the multiplier.

   Table 1

   Let X = 1100 (multiplier) Let Y = 0010 (multiplicand)

   2s complement of Y; Y = 1110 Load the X value in the table.

   Load 0 for X-1 value it should be the previous first least significant bit of X .

   Load 0 in U and V rows which will have the product of X and Y at the end of operation.

   Make four rows for each cycle; this is because we are multiplying four bits numbers.

   Step 2: Booth Algorithm

   Booth algorithm requires examination of the multiplier bits, and shifting of the partial product. Prior to the shifting, the multiplicand may be added to partial product, subtracted from the partial product, or left unchanged according to the following rules:

   Table 2.

   Look at the first least significant bits of the multiplier X, and the previous least significant bits of the multiplier X – 1.

   1. 0 Shift only

   2. 1 Shift only.

   1. 1 Add Y to U, and shift

   2. 0 Subtract Y from U, and shift or add (-Y) to U and shift.

   Take U & V together and shift arithmetic right shift which preserves the sign bit of 2s complement number. Thus a positive number remains positive, and a negative number remains negative. Shift X circular right shift because this will prevent us from using two registers for the X value. Repeat the same steps until the four cycles are completed. [8]

   bits of the multiplier. [7] Multiplier is equal to 0 1 0 1 1 10 then a 0 is placed to the right most bit which gives

   0 1 0 1 1 10 0 the 3 digits are selected at a time with overlapping left most bit as follows:

   Figure 8 Grouping of three bits

   Table 4. Encoding of Radix-4 Booth Multiplier[9] [10] [11]

   Table 3.

2. Radix-4

Radix-4 Booth algorithm scans strings of 3 bits with the algorithm given below Append a 0 to the right side of the LSB of the multiplier consider the bits in groups of three, in a way that each group overlaps with the previous group by one bit. Grouping starts from the LSB and the first group only uses 2 bits of the multiplier. According to the value of each vector, Partial Product will be 0, +Y, Y, +2Y,2Y. The negative values of y are considered by taking the 2s complement to the Booth recode the multiplier term, we have to consider the bits in groups of three, in a way that each group overlaps with the previous group by one bit. Grouping starts from the LSB and the first group only uses 2

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VII. CONCLUSION

We found that the parallel multipliers are much faster than the serial multiplier. In case of parallel multipliers, the total area is much less than that of serial multipliers. Hence the power consumption is also less. This speeds up the calculation and makes the system faster. While comparing the radix 2 and the radix 4 booth multipliers we found that radix 4 Consumes lesser power than that of radix 2. This is because it uses almost half number of iteration and adders when compared to radix 2. When all the multipliers were compared we found that array multipliers are most power consuming and have the maximum area. This is because it uses a large number of adders. As a result it slows down the system because now the system has to do a lot of calculation. Multipliers are one the most important component of many systems. So we always need to find a better solution in case of multipliers. Our multipliers should

always consume less power and cover less area. In the end we determine that radix 4 modified booth algorithm works the best.

REFERENCES

Booth 27s Algorithm Calculator Download

1. Prasanna Raj P, Rao, Ravi, VLSI Design and Analysis of Multipliers For Low Power, Intelligent Information Hiding and Multimedia Signal

   Processing, Fifth International Conference, pp.: 1354-1357, Sept. 2009.

2. R.I.Hartle, K.K.Pardhi, Digit-Serial Computation,Kluwer Academic,Boston,MA 1995

3. Jayashree Taralabenchi, Kavana Hegde, Soumya Hegde,

   Implementation of Binary Multiplication using Booth and Systolic Algorithm on FPGA using VHDL, International Conference & Workshop on Recent Trends in Technology, (TCET) 2012 Proceedings published in International Journal of Computer Applications® (IJCA) .

4. Louis P. Rubinfield, A Proof of the Modified Booth's Algorithm for Multiplication, Computers, IEEE Transactions,vol.24, pp.: 1014- 1015, Oct. 1975

5. Depth:In More Booths Algorithm, staff.ustc.edu.cn/~han/CS152CD/Content/COD3e/inmoredepth/IMD3

   -Booths-Algorithm.pdf – –

6. Abenet Getahun, Booth Multiplication Algorithm, Fall 2003 CSCI 401

7. Sandeep Shrivastava*, Jaikaran Singh* and Mukesh Tiwari*,

   Implementation of Radix-2 Booth Multiplier and Comparison with Radix-4 Encoder Booth Multiplier, International Journal on Emerging Technologies 2(1): 14-16(2011) ISSN : 0975-8364

8. Abenet Getahun, Booth Multiplication Algorithm, Fall 2003 CSCI 401

9. Fabrizio Lamberti, Nikolaos Andrikos,Elisardo Antelo, Paolo Montuschi, Speeding-up Booth Encoded Multipliers by Reducing the Size of Partial Product Array,' Internal Report Dauin/Delen- Politecnico Di Torino and University De Santiago De Compostela, 2009

10. S. Shafiulla Basha1, Syed. Jahangir Badashap, Design and Implementation of Radix-4 Based High Speed Multiplier For ALUS Using Minimal Partial Products,International Journal of Advances in Engineering & Technology, July 2012. ©IJAET ISSN: 2231-1963

11. H. S. Krishnaprasad Puttam1, P. Sivadurga Rao2 & N. V. G. Prasad3,

    Implementation of Low Power and High Speed Multiplier- Accumulator Using SPST Adder and Verilog,International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3390-3397 ISSN: 2249-6645.

12. J.Umamageshwari M.Yeni Saranya M.tech, Asic implementation of low power high radix booth encoded multiplier using spst, international journal of communications and engineering volume 05 no.5, issue: 02 marcp012 .

13. Stephen Brown Zvonko Vranesic, Fundamentals of Digital Logic Degin with VHDL, Tata mcgraw-Hill