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Booth's Multiplication Algorithm

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 2k to weight 2m that can be considered as 2k+ 1 - 2m.

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

Booth's Multiplication Algorithm

In the above flowchart, initially, AC and Qn + 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 Qn and Qn + 1, where Qn represents the last bit of QR, and Qn + 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 Qn + 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 Qn + 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 Qn+1 shows the incremented bit of Qn by 1.
  5. On each cycle of the booth algorithm, Qn and Qn + 1 bits will be checked on the following parameters as follows:
    1. When two bits Qn and Qn + 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 Qn + 1 is incremented by 1 bit.
    2. If the bits of Qn and Qn + 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 Qn and Qn + 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.

Booth's Multiplication Algorithm

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.

Qn Qn + 1 M = (0111)
M' + 1 = (1001) & Operation
AC Q Qn + 1 SC
1 0 Initial 0000 0011 0 4
Subtract (M' + 1) 1001
1001
Perform Arithmetic Right Shift operations (ashr) 1100 1001 1 3
1 1 Perform Arithmetic Right Shift operations (ashr) 1110 0100 1 2
0 1 Addition (A + M) 0111
0101 0100
Perform Arithmetic right shift operation 0010 1010 0 1
0 0 Perform Arithmetic right shift operation 0001 0101 0 0

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)

Qn             Qn + 1 M = 0 1 0 1 1 1
M' + 1 = 1 0 1 0 0 1
AC Q Qn + 1             SC
Initially 000000 110111 0             6
1             0 Subtract M 101001
101001
Perform Arithmetic right shift operation 110100 111011 1             5
1             1 Perform Arithmetic right shift operation 111010 011101 1             4
1             1 Perform Arithmetic right shift operation 111101 001110 1             3
0              1 Addition (A + M) 010111
010100
Perform Arithmetic right shift operation 001010 000111 0             2
1             0 Subtract M 101001
110011
Perform Arithmetic right shift operation 111001 100011 1             1
1             1 Perform Arithmetic right shift operation 111100 110001 1             0

Qn + 1 = 1, it means the output is negative.

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


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