Question

The following two signed 2’s complement numbers (multiplicand \( M \) and multiplier \( Q \)) are being multiplied using Booth’s algorithm: 

The total number of addition and subtraction operations to be performed is ___________. (Answer in integer)

Hint:

Booth's multiplication algorithm reduces the number of addition/subtraction operations by encoding runs of 1s efficiently. The number of operations depends on the bit pattern of the multiplier.

Answer 1

Booth’s multiplication algorithm works by examining pairs of bits in the multiplier and deciding whether to add, subtract, or shift based on the pattern. The key steps include:

• Scanning the bits of \( Q \) from right to left: This step is essential in Booth's algorithm as the multiplier's bits are examined in pairs for pattern recognition.
• Performing addition when encountering a '01' transition: When the bit pair '01' is found, it indicates that the result should be increased, so an addition operation is performed.
• Performing subtraction when encountering a '10' transition: When the bit pair '10' is encountered, it signals a need to subtract, and so a subtraction operation is executed.
• Shifting in all other cases: If the bit pair is '00' or '11', a shift operation is performed to continue the process without modifying the result.

For the given numbers \( M = 1100 1101 1110 1101 \) and \( Q = 1010 0100 1010 1010 \), applying Booth’s algorithm results in 13 addition and subtraction operations. This is the total number of operations performed to compute the product using the algorithm, based on the transitions between bit pairs in the multiplier \( Q \).