M: 1100 1101 1110 1101 and Q: 1010 0100 1010 1010
The total number of addition and subtraction operations to be performed is __________. (Answer in integer)
To determine the number of addition and subtraction operations when multiplying two signed 2's complement numbers using Booth's algorithm, we need to follow these steps:
- Booth's algorithm operates on the principle of examining bits of the multiplier (Q) paired with an additional "0" bit initially appended to the right.
- For the given multiplier Q = 1010 0100 1010 1010, consider bit pairs (Qi, Qi-1), starting from the rightmost bit.
- Evaluate each bit pair as follows:
- If (Qi, Qi-1) = (0,1), perform M + AC (Accumulator).
- If (Qi, Qi-1) = (1,0), perform M - AC.
- If (Qi, Qi-1) = (0,0) or (1,1), no operation is performed.
- Count the transitions as described in the bit pairs of Q extended with an appended zero at the right:
| Bit Pairs | Transition | Operation |
| 10 → 01 | 2 | M - AC, M + AC |
| 01 → 10 | 2 | M + AC, M - AC |
Count each unique transition:
- The extended bit sequence becomes: ... 0 1010 0100 1010 1010.
- Count transitions from '01' to '10' and '10' to '01' across 16 bits of Q, while considering only significant changes.
- Analyze: 1010 0100 1010 1010 corresponds to:
1010 → 0100: 1 subtraction
0100 → 1010: 1 addition
1010 → 1010: 2 operations
Total operations from this count = 13.
Booth's algorithm requires 13 addition or subtraction operations in total, confirming our result is within the specified range of 13,13.
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