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ECE 352 Digital System Fundamentals

Learn how to optimize the slow part of carry look-ahead adders by separating carry and sum logic, using partial full adders, and modifying the carry-chain logic. See how carry look-ahead logic reduces area and increases speed compared to ripple-carry adders.

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ECE 352 Digital System Fundamentals

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  1. ECE 352Digital System Fundamentals Carry Look-Ahead Adders

  2. Optimize The Slow Part: Carry Chain • For big adders, the carry chain is very long… • Separate the carry and sum logic in a full adder so that we can optimize the carry logic • Partial Full Adder (PFA): a full adder without carry logic • Modify the carry-chain logic to be a 2-level function instead of a multi-level function Carry Chain PFA

  3. Carry Look-Ahead Adder (CLA) • As usual, can trade area/power for speed • Multi-level logic (ripple carry) reduces area • Flattening carry logic increases speed • Make carry logic a 2-level function of: • Generate (G) • “Based on the operand values at just this bit position, will the carry-out for this position be 1 regardless of the carry-in?” • Propagate (P) • “Based on the operand values at just this bit position, will the carry-out for this position be equal to the carry-in?”

  4. Ripple-Carry Adder • Each carry bit is a multi-level function of the generates and propagates of the lower positions Ripple-carryadder C3 equation C3 = G2 + P2 C2 = G2 + P2 (G1 + P1 C1) = G2 + P2 (G1 + P1 (G0 + P0 C0))

  5. Carry Look-Ahead Adder • Each carry bit is a two-level function of the generates and propagates of the lower positions

  6. Getting There Algebraically • “Flatten” the ripple-carry equation to two levels C3 = G2+ P2 (G1 + P1 (G0 + P0 C0)) = G2 + P2G1+ P2P1 (G0 + P0 C0) = G2 + P2 G1 + P2 P1G0+ P2 P1P0C0 Carry look-ahead adder C3 equation

  7. Carry Look-Ahead Logic • You do not need to memorizethe function for each carry bit! • Instead, remember whatgenerate and propagatemean, and how MATH works C3 C0 A3 A2 A1 A0 + B3 B2 B1 B0 P2 P2 G2 + G1 P2 C3 = P1 P0 + C0 P1 + G0 bit 2 generates a carry, C3 will be 1 if and it propagates at bit 2, or if bit 1 generates a carry or if bit 0 generates a carry and bit 2, and it propagates at bit 1 or if the carry-in at the least-significant bit is 1 and bit 1 and it propagates at bit 0 and bit 2.

  8. Extending Carry Look-Ahead • Higher carry indices require more AND gates and a bigger OR gate • There is a practical limit on the size of gates… • How can we make wider adders? Hierarchy! • Add look-ahead outputs indicating if the 4-bit group of PFAs generate or propagate a carry to the next group • A higher-level look-ahead block uses group generates and propagates to determine carry-ins for each group • Same idea as before, but for a group of bits • “Does this GROUP generate a carry?” • “Does this GROUP propagate a carry?”

  9. Extending Carry Look-Ahead • Replace C4 output with a group generate output GG, and a group propagate output PG GG is 1 if the group will generate a carry PG is 1 if the group will propagate the carry

  10. 16-Bit Carry Look-Ahead Adder • Entire 4-bit carry look-ahead adders (instead of PFAs) are connected to a top-level look-ahead • Top-level look-ahead produces carry-in bits for each 4-bit CLA block based on group generates and propagates of the lower CLA blocks 4-bit look-ahead block 4-bit carry look-ahead adders

  11. 16-Bit Carry Look-Ahead Adder • Entire 4-bit carry look-ahead adders (instead of PFAs) are connected to a top-level look-ahead • Top-level look-ahead produces carry-in bits for each 4-bit CLA block based on group generates and propagates of the lower CLA blocks 4-bit look-ahead block C4 = GG0 + PG0C0

  12. 16-Bit Carry Look-Ahead Adder • Entire 4-bit carry look-ahead adders (instead of PFAs) are connected to a top-level look-ahead • Top-level look-ahead produces carry-in bits for each 4-bit CLA block based on group generates and propagates of the lower CLA blocks 4-bit look-ahead block C12 = GG2 + PG2GG1 + PG2PG1GG0 + PG2PG1PG0C0

  13. Extending Further… • We can make even larger carry look-ahead adders by adding more levels of hierarchy • Build a 64-bit CLA with four 16-bit CLAs plus another look-ahead block • Each 16-bit CLA would need a group generate and a group propagate output in its top-level look-ahead • Does this group of 16 bits generate a carry? • Does this group of 16 bits propagate a carry? • Hierarchical CLA carry chain is no longer two-level • But far fewer levels than if it were a ripple-carry adder! • CLA delay grows more slowly with operand size

  14. ECE 352Digital System Fundamentals Carry Look-Ahead Adders

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