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VLSI Digital System Design

VLSI Digital System Design

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VLSI Digital System Design

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  1. Multiplication VLSI Digital System Design

  2. Booth's Algorithm • Multiplication can be sped up • When large number of consecutive 1sin multiplier • Replace consecutive additions • Subtract at least-significant end • Add at position to the left ofthe most-significant end

  3. 16-Bit Booth's Algorithm Examples • Bits in multiplier andcorresponding actions to take • 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 01 0 0 0 -1 0 0 1 -1 1 0 -1 1 -1 1 -1 0 • 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 01 -1 0 1 0 0 -1 1 0 -1 1 -1 1 0 0 -1 0

  4. Higher-Radix Multiplication • Retire k low-order bits of multiplierat each step, rather than just one bit: radix 2 • E.g., let k = 2: radix 4 • Let b = multiplicand • If 00, add 0 • If 01, add b • If 10, add 2b: shift and add • If 11, add 2b + b: avoid with Booth recoding

  5. Booth Recoding • If 11, add 2b + b: avoid with Booth recoding • 0011 = 0100 – 0001 • If 11, subtract b andadd 1 to next-higher group of k bits • Also called overlapping triplets • It looks at 3 bits at a time • Ordinary radix-4 multiplication looks at2 bits at a time

  6. Radix-4 Booth Recoding Rules • Cur pair Previous bit i+1 i i–1 Add 2i+1 Add 2i Add0 0 0 00 00 +00 0 1 00 01 End +b +b 0 1 0 01 End +2b 10 Beg -b +b 0 1 1 01 End +2b 11 Cont +2b1 0 0 10 Beg -2b 00 -2b1 0 1 10 Beg -2b 01 End +b -b1 1 0 11 Cont 10 Beg -b -b1 1 1 11 Cont 11 Cont +0

  7. Booth Recoding Control Signals • Cur pair Previous bit i+1 i i–1 Add Neg Zero Two0 0 0 +0 0 1 00 0 1 +b 0 0 00 1 0 +b 0 0 00 1 1 +2b 0 0 11 0 0 -2b 1 0 11 0 1 -b 1 0 01 1 0 -b 1 0 01 1 1 +0 1 1 0