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Booth Multiplier

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• 1. Multiplication
• 2. Multiplier NotationPartial ProductsLogical-AND
• 4. Shift and Add Examples
• 5. Programmed Multiplication
• 6. Programmed Multiplication (cont.)
• 7. Hardware Shift and Add (right)
• 8. Hardware Shift and Add
• 9. Hardware Shift and Add (left)
• 10. Signed Number Multiplication (positive case)
• 11. Signed Number Multiplication (negative case)
• 12. Booth’s Recoding (or encoding)• Developed for Speeding Up Multiplication in Early Computers• When a Partial Product of 0 Occurs, Can Skip Addition and Just Shift• Doesn’t Help Multipliers Where Datapaths Go Through Adder Such as Previous Examples• Does Help Designs for Asynchronous Implementation or Microprogramming Since Shifting is Faster Than Addition• Variable Delay – Depends on Number of One’s in• Booth Observed that a String of 1’s May be Replaced as: j −1 i +1 j +1 2 +2 j +L+ 2 +2 =2 i −2 i
• 13. Booth’s Recoding Examplexn xn-1 ... xi xi-1 ... x0 (0) yi=xi-1 - xi yn ... yi ... y0 xi xi-1 Operation Comments yi 0 0 shift only string of zeros 0 1 1 shift only string of ones 0 1 0 subtract shift beg. string of ones -1 0 1 addition shift end string of ones 1EXAMPLE 0011110011(0) 0100010101
• 14. Booth’s Recoding• Maps Words With Digit Set [0,1] to Those With [-1,1]
• 15. Sequential Multiplication A 1011 (-510) X 1101 (-310) Y 0111 (recoded)(-1) Add –A 0101Shift 00101(+1) Add +A 1011 11011Shift 111011(-1) Add –A 0101 001111Shift 0001111 (+1510)
• 16. Booth Multiplier Example
• 17. Booth’s Recoding Drawbacks• Number of add/sub Operations are Variable• Some Inefficiencies EXAMPLE 001010101(0) 011111111• Can Use Modified Booth’s Recoding to Prevent• Will Look at This in Later Class
• 18. Sign Extension• Consider 6-bit 2’s Complement Number s=0 Positive Value; s=1 Negative Value• Show Sign Extension Works:s s s s s p4 p3 p2 p1 p0= − s ×29 + s ×28 + s ×27 + s ×26 + s ×25 + p4 ×24 + p3 ×23 + p2 ×2 2 + p1 ×21 + p0 ×20 4= − s ×2 + s ×(2 + 2 + 2 + 2 ) + ∑ pi ×2i 9 8 7 6 5 i =0 4= − s ×2 + s ×(2 − 2 ) + ∑ pi ×2i 9 9 5 i =0 4= − s ×2 + ∑ pi ×2i 5 i =0 • Definition of 2’s Complement
• 19. Sign Extension Example A 010110 (+2210) X 001011 (+1110) Y 010101 (recoding) 11111101010 (neg. A) 0000000000 (0 A) 111101010 (neg. A) 00000000 (0 A) 0010110 (neg. A) 000000 (0 A) 00011110010 (24210)
• 20. Sign Extension Example• Same Trick as Before, Complement Original Sign Bit• Add 1 to Column 5 1 001010 (neg. A) 100000 (0 A) 001010 (neg. A) 100000 (0 A) 110110 (neg. A) 100000 (0 A) 00011110010 (24210)
• 21. Methods for Fast Multiplication• Reduce Number of Partial Products to be Added – Group Multiplier Bits Together – Higher Radix Multiplier• Add the Partial Products Faster