Lookahead carry unit

Last updated June 02, 2023

A lookahead carry unit (LCU) is a logical unit in digital circuit design used to decrease calculation time in adder units and used in conjunction with carry look-ahead adders (CLAs).

4-bit adder

A single 4-bit CLA is shown below:

16-bit adder

By combining four 4-bit CLAs, a 16-bit adder can be created but additional logic is needed in the form of an LCU.

The LCU accepts the group propagate ( $P_{G}$ ) and group generate ( $G_{G}$ ) from each of the four CLAs. $P_{G}$ and $G_{G}$ have the following expressions for each CLA adder:^[1]

P_{G}=P_{0}\cdot P_{1}\cdot P_{2}\cdot P_{3}

G_{G}=G_{3}+G_{2}\cdot P_{3}+G_{1}\cdot P_{2}\cdot P_{3}+G_{0}\cdot P_{1}\cdot P_{2}\cdot P_{3}

The LCU then generates the carry input for each CLA.

Assume that $P_{i}$ is $P_{G}$ and $G_{i}$ is $G_{G}$ from the i^th CLA then the output carry bits are

C_{4}=G_{0}+P_{0}\cdot C_{0}

C_{8}=G_{4}+P_{4}\cdot C_{4}

C_{12}=G_{8}+P_{8}\cdot C_{8}

C_{16}=G_{12}+P_{12}\cdot C_{12}

Substituting $C_{4}$ into $C_{8}$ , then $C_{8}$ into $C_{12}$ , then $C_{12}$ into $C_{16}$ yields the expanded equations:

C_{4}=G_{0}+P_{0}\cdot C_{0}

C_{8}=G_{4}+G_{0}\cdot P_{4}+C_{0}\cdot P_{0}\cdot P_{4}

C_{12}=G_{8}+G_{4}\cdot P_{8}+G_{0}\cdot P_{4}\cdot P_{8}+C_{0}\cdot P_{0}\cdot P_{4}\cdot P_{8}

C_{16}=G_{12}+G_{8}\cdot P_{12}+G_{4}\cdot P_{8}\cdot P_{12}+G_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}+C_{0}\cdot P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}

$C_{4}$ corresponds to the carry input into the second CLA; $C_{8}$ to the third CLA; $C_{12}$ to the fourth CLA; and $C_{16}$ to overflow carry bit.

In addition, the LCU can calculate its own propagate and generate:

P_{LCU}=P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}

G_{LCU}=G_{12}+G_{8}\cdot P_{12}+G_{4}\cdot P_{8}\cdot P_{12}+G_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}

C_{16}=G_{LCU}+C_{0}\cdot P_{LCU}

64-bit adder

By combining 4 CLAs and an LCU together creates a 16-bit adder. Four of these units can be combined to form a 64-bit adder. An additional (second-level) LCU is needed that accepts the propagate ( $P_{LCU}$ ) and generate ( $G_{LCU}$ ) from each LCU and the four carry outputs generated by the second-level LCU are fed into the first-level LCUs.

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References

↑ "Carry Look Ahead Adder". Archived from the original on 2011-09-25. Retrieved 2011-10-07.

Katz, Randy (1994). Contemporary Logic Design . The Benjamin/Cummings Publishing Company. pp. 249–256. ISBN 0-8053-2703-7.
Vahid, Frank (2006). Digital Design . John Wiley and Sons Publishers. pp. 296–316. ISBN 0-470-04437-3.

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[1] "Carry Look Ahead Adder". Archived from the original on 2011-09-25. Retrieved 2011-10-07.

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