Lookahead carry unit

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_3 \cdot P_2 \cdot P_1 \cdot P_0

G_G = G_3 + P_3 \cdot G_2 + P_3 \cdot P_2 \cdot G_1 + P_3 \cdot P_2 \cdot P_1 \cdot G_0

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 propragate ( $P_{LCU}$ ) 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.

References

↑ http://www.seas.upenn.edu/~ese171/lab/CarryLookAhead/CarryLookAheadF01.html

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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