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Unrolling Carry Recurrence

Unrolling Carry Recurrence

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Unrolling Carry Recurrence

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  1. Unrolling Carry Recurrence

  2. Carry-lookahead Equations

  3. 4-bit CLA

  4. Circuit Structure

  5. CLA Complexity

  6. Managing CLA Complexity

  7. Example CLA

  8. Radix-16 Addition • Two Binary Numbers Grouped into Hex Digits • Block Generate and Propagate Signals in Each radix-16 Digit • Replace c4 Position of CLA Network with Block Signals g[i,i+3] and p[i,i+3] • Results in 4-bit “Lookahead Carry Generator”

  9. CLA Design

  10. Lookahead Carry Generator

  11. Block Generate and Propagate • Assume i0 < i1 < i2 • Example: g[0,3] is Generate Signal of Block for bits 0-3 • Relationships Allow for Merging of Blocks • Can Allow Merged Block to Overlap

  12. Example Merged LAG x15-12 y15-12 x11-8 y11-8 x7-4 y7-4 x3-0 y3-0 CLA 3 CLA 2 CLA 1 CLA 0 c12 c8 c4 c0 g15-12 p15-12 g11-8 p11-8 g7-4 p7-4 g3-0 p3-0 s15-12 s11-8 s7-4 s3-0 Lookahead Carry Generator g15-0 p15-0

  13. CLA Latency

  14. CLA Architecture

  15. Overlapped LAGs • Overlap Blocks [i1,j1] and [i0,j0] • Relationships Become: • Useful for Building Trees of Different Shapes

  16. CLA With LAG

  17. CLA Latency

  18. Ling Adders

  19. Ling Adders – Wired OR